Problems and Solutions - University of Johannesburgissc.uj.ac.za/downloads/problems/analysis.pdf · Problems and Solutions in Real and Complex Analysis, Integration, Functional Equations

Problems and SolutionsinReal and Complex Analysis,Integration,Functional EquationsandInequalities

byWilli-Hans SteebInternational School for Scientific ComputingatUniversity of Johannesburg, South Africa

Preface

The purpose of this book is to supply a collection of problems in analysis.Please submit your solution to one of th email addresses below.

e-mail addresses of the author:

[email protected][email protected]

Home page of the author:

http://issc.uj.ac.za

Prescribed books for problems.

1) Problems and Solutions in Theoretical and Mathematical Physics, ThirdEdition, Volume I: Introductory Level

by

Willi-Hans SteebWorld Scientific Publishing, Singapore 2009ISBN- 13 978 981 4282 14 7http://www.worldscibooks.com/physics/7416.html

2) Problems and Solutions in Theoretical and Mathematical Physics, ThirdEdition, Volume II: Advanced Level

by

Willi-Hans SteebWorld Scientific Publishing, Singapore 2009ISBN-13 978 981 4282 16 1http://www.worldscibooks.com/physics/7416.html

updated: January 16, 2018

v

http://www.worldscibooks.com/physics/7416.html

http://www.worldscibooks.com/physics/7416.html

vi

Contents

Preface v

Notation x

1 Sums and Products 11.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 12

2 Maps 262.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 35

3 Functions 413.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 60

4 Polynomial 734.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 86

5 Equations 915.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 98

6 Normed Spaces 1026.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 1036.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 106

7 Complex Numbers and Complex Functions 1077.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 1077.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 113

vii

8 Integration 1188.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 1188.2 Programming Problems . . . . . . . . . . . . . . . . . . . . 1448.3 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 145

9 Functional Equations 1529.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 1529.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 156

10 Inequalities 15710.1 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . 15710.2 Supplementary Problems . . . . . . . . . . . . . . . . . . . . 169

Bibliography 172

Index 177

viii

x

Notation

:= is defined as∈ belongs to (a set)/∈ does not belong to (a set)∩ intersection of sets∪ union of sets∅ empty setN set of natural numbersZ set of integersQ set of rational numbersR set of real numbersR+ set of nonnegative real numbersC set of complex numbersRn n-dimensional Euclidean space

space of column vectors with n real componentsCn n-dimensional complex linear space

space of column vectors with n complex componentsH Hilbert spacei

√−1

<z real part of the complex number z=z imaginary part of the complex number z|z| modulus of complex number z

|x+ iy| = (x2 + y2)1/2, x, y ∈ RT ⊂ S subset T of set SS ∩ T the intersection of the sets S and TS ∪ T the union of the sets S and Tf(S) image of set S under mapping ff g composition of two mappings (f g)(x) = f(g(x))v column vector in CnvT transpose of v (row vector)0 zero (column) vector‖ . ‖ normx · y ≡ x∗y scalar product (inner product) in Cnx× y vector product in R3

A,B,C m× n matricesdet(A) determinant of a square matrix Atr(A) trace of a square matrix Arank(A) rank of matrix AAT transpose of matrix A

xi

A conjugate of matrix AA∗ conjugate transpose of matrix AA† conjugate transpose of matrix A

(notation used in physics)A−1 inverse of square matrix A (if it exists)In n× n unit matrixI unit operator0n n× n zero matrixAB matrix product of m× n matrix A

and n× p matrix BA •B Hadamard product (entry-wise product)

of m× n matrices A and B[A,B] := AB −BA commutator for square matrices A and B[A,B]+ := AB +BA anticommutator for square matrices A and BA⊗B Kronecker product of matrices A and BA⊕B Direct sum of matrices A and Bδjk Kronecker delta with δjk = 1 for j = k

and δjk = 0 for j 6= kλ eigenvalueε real parametert time variableH Hamilton operator

Chapter 1

Sums and Products

1.1 Solved ProblemsProblem 1. The harmonic series can be approximated by

n∑j=1

1j≈ 0.5772 + ln(n) +

12n.

Calculate the left and rigt-hand side for n = 1 and n = 10.

Problem 2. The Bernoulli numbers B0, B1, B2, . . . are defined by thepower series expansion

x

ex − 1=∞∑j=0

Bjj!xj ≡ B0 +

B1

1!x+

B2

2!x2 + · · · .

One finds B0 = 1, B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30. One hasBj = 0 if j ≥ 3 and j odd. The Bernoulli numbers are utilized in the Eulersummation formulan∑j=1

f(j) =∫ n

1

f(t)dt+12

(f(n)+f(1))+n∑k=1

B2k

(2k)!(f (2k−1)(n)−f (2k−1)(1))+Rm(n)

where|Rm(n)| ≤ 4

(2π)2m

∫ m

1

|f (2m)(t)|dt.

1

2 Problems and Solutions

Calculate the sumn∑j=1

j2

using f(t) = t2.

Problem 3. Let z ∈ C and |z| < 1. Calculate the sum

(1− |z|2)2s∞∑n=0

n

(2s+ n− 1

n

)|z|2n.

Problem 4. Let x ∈ R and r ∈ N with r ≥ 1. Find the sum

r−1∑k=0

exp(−2πikx/r).

Problem 5. Let a1, a2, . . . , an ∈ [0, 1]. Show that there exists a real num-ber x in the unit interval such that the average of the (unsigned) distancesfrom x to the aj ’s is 1/2.

Problem 6. Let f : Rm → R be a differentiable function. Show thatthere exist m differentiable functions g1, g2, . . . , gm defined on Rm withthe properties

f(x) = f(0) +m∑j=1

xjgj(x)

and

gj(0) =∂f

∂xj(0)

where x = (x1, x2, . . . , xm).

Problem 7. Let n be a positive integer and f(j) = j(j − 1)(j − 2) withj = 1, 2, . . . , n. Let aj := f(j + 1)− f(j). By calculating the sum

∑nj=1 aj

show thatn∑j=1

j2 =n(n+ 1)(2n+ 1)

6.

Sums and Products 3

Problem 8. Show that N∑j=1

1u− vj

2

−N∑j=1

1(u− vj)2

≡ 2N∑i<jj=2i=1

1vi − vj

(1

u− vi− 1u− vj

).

This identity plays a role in the Bethe ansatz.

Problem 9. Show that the series

f(θ) =∞∑j=0

sin(3jθ)2j

is convergent. Is the series df/dθ convergent?

Problem 10. Find the radius of convergence of the power series

∞∑j=0

(j + k

j

)zj , k > 0.

Problem 11. Find the radius of convergence of the power series

∞∑j=1

j!jjzmj , m = 1, 2, . . . .

Problem 12. Let (s0, s1, ..., sn−1)T ∈ Rn, where n = 2k. This vector inRn can be associated with a piecewise constant function f defined on [0, 1)

f(x) =2k−1∑j=0

sjΘ[j2−k,(j+1)2−k)(x)

where Θ[j2−k,(j+1)2−k)(x) is the step function

Θ[j2−k,(j+1)2−k)(x) :=

1 x ∈ [j2−k, (j + 1)2−k)0 x /∈ [j2−k, (j + 1)2−k)

with the support [j2−k, (j + 1)2−k). Let xj+1 = 4xj(1 − xj) with j =0, 1, 2, . . . and x0 = 1/3. Then

x0 =13, x1 =

89, x2 =

3281, x3 =

62726561

.


Find f(x) for this data set and then calculate∫ 1

0

f(x)dx.

Problem 13. Let a1, a2, . . . , an be a finite sequence of numbers. ItsCesaro sum is defined as

s1 + s2 + · · ·+ snn

wheresk = a1 + a2 + · · ·+ ak

for each k, 1 ≤ k ≤ n. Suppose that the Cesaro sum of the 99-term sequencea1, a2, . . . , a99 is 100. Find the Cesaro sum of the 100-term sequence 1,a1, a2, . . . , a99.

Problem 14. Each x ∈ [0, 1] can be written as

x =∞∑j=1

εj2j

with εj = 0 or εj = 1. Define the function f : [0, 1]→ [0, 1) as

f(x) =∞∑j=1

2εj3j.

The function f is known as Cantor function. Let x = 1/8. Find f(x).

Problem 15. Let s ≥ 2. Simplify the sum

∞∑j1=1

∞∑j2=1

1(j1 + j2)s

.

Problem 16. Consider

1 + x

1− x− x2=∞∑j=0

cjxj .

Find cj .

Sums and Products 5

Problem 17. The Cantor series approximation is defined as follows.For arbitrary chosen integers n1, n2, . . . (equal or larger than 2), we canapproximate any real number r0 as follows

xj = integer part(rj), j = 0, 1, 2, . . .rj+1 = (rj − xj)nj

and

r0 ≈ x0 +N∑j=1

xjn1n2 · · ·nj

.

The approximation error is

EN =1

n1n2 · · ·nN.

Apply this approximation to r0 = 2/3 and the golden mean number withnj = 2 for all j and N = 4.

Problem 18. Calculate the sum

S =√

2 +√

2−√

2√

2−√

2−√

2−√

2.

Problem 19. Let n1, n2, n3 ∈ Z. Calculate

1− eiπ(n1+n2+n3).

Problem 20. Show that 7 + 2√

10 has a square root the form√x+√y.

Problem 21. Calculate∞∑j=0

j3 + 1j!

xj .

Hint. Write j3 + 1 in the form

a+ bj + cj(j − 1) + dj(j − 1)(j − 2).

Problem 22. Let n ∈ N. Find the sum

(n−1)/2∑k=−(n−1)/2

k2


where k runs in steps of 1.

Problem 23. Assume that the series

1, 14, 51, 124, 245, 426, . . .

is of the formak3 + bk2 + ck + d.

Find the integer coefficents a, b, c, d.

Problem 24. Find the sum

Sn =2

1 · 2 · 3+

22 · 3 · 4

+2

3 · 4 · 5+ · · ·+ 2

n · (n+ 1) · (n+ 2).

Calculate Sn for n→∞. Hint. Find a, b, c for

2n · (n+ 1) · (n+ 2)

=a

n+

b

n+ 1+

c

n+ 2.

Problem 25. Let a > 0. Show that

∞∑n=−∞

1z − na

=1z

+ 2z∞∑n=1

1z2 − n2a2

.

Problem 26. Let x 6= y, x 6= z, y 6= z. Find the sum

1(x− y)(x− z)

+1

(y − x)(y − z)+

1(z − x)(z − y)

.

Problem 27. Given the expansion

ln(1 + x) = x− x2

2+x3

3− x4

4+ · · · , −1 < x ≤ 1.

Calculate ln(11).

Problem 28. Let n ∈ N0 and p, a > 0. Show that

∞∑k=0

(k + n)!k!n!

exp(−2πkp/a) ≡ 1(1− exp(−2πp/a))n+1

.

Sums and Products 7

Hint: Start with

f(n) :=∞∑k=0

(k + n)!k!n!

exp(−2πkp/a)

and find f(n+ 1).

Problem 29. Let (s1, s2) ∈ Z2. Let P (φ1, φ2) be a probability densityand let θ1, θ2 ∈ R. We can express the characteristic double sequence as

χ(s1, s2) =∫ θ1+2π

θ1

∫ θ2+2π

θ2

exp(i(s1φ1 + s2φ2))P (φ1, φ2)dφ1dφ2.

Find P (φ1, φ2) as a function of χ(s1, s2). Note that χ(0, 0) = 1.

Problem 30. Show thatπ

4= 4 arctan(1/5)− arctan(1/239).

Hint. Let θ = arctan(1/5). Thus tan(θ) = 1/5. Applying the double angleformula for the tangent we have tan(2θ) = 5/12 and tan(4θ) = 120/119.Finally apply that tan(π/4) = 1.

Problem 31. Let n ∈ N0. Consider an infinite number of time variablet = (t1, t2, t3, . . .). Consider the sum

pn(x+ t1, t2, t3, . . .) =∑

k0+k1+2k2+3k3+···=nk0,k1,k2,k3,...≥0

xk0tk11 tk22 · · ·

k0!k1!k2! · · ·.

Find p0, p1, p2, p3.

Problem 32. Let n be an integer with n ≥ 1. Simplify the series

S =1

1 · 2+

12 · 3

+1

3 · 4+ · · ·+ 1

n(n+ 1).

Problem 33. Let N be a positive integer and a, b be positive integers orpositive half-integers. Find

f(N, a, b) =12

(1 + (−1)2a + (−1)2b + (−1)2a+2b+N ).

Problem 34. Let −1 < x < +1 and n ≥ 2.


(i) Show that

Sn(x) =n−1∑j=0

jxj =nxn

x− 1+x(1− xn)(1− x)2

.

(ii) Show thatlimn→∞

Sn(x) =x

(1− x)2.

(iii) Let x = 1/2. Show that limn→∞ Sn(1/2) = 2.

Problem 35. Let x, y, z ∈ R and x 6= y, x 6= z, y 6= z. Show that

x

(z − x)(x− y)+

y

(x− y)(y − z)+

z

(y − z)(z − x)= 0.

Problem 36. Let

x 6= 0, x 6= ±2π, x 6= ±4π, . . .

and n ∈ N. Show that

sin(x) + sin(2x) + · · ·+ sin(nx) =cos(x/2)− cos((n+ 1/2)x)

2 sin(x/2).

For the values x = 0, x = ±2π, x = ±4π etc the sum is given by 0.

Problem 37. Let n = 0, 1, 2, . . .. The function

Kn(θ) :=1

n+ 1sin2

((n+1

2

)θ)

sin2(

12θ)

is called the Fejer kernel.(i) Show that

Kn(θ) :=j=+n∑j=−n

(1− |j|

n+ 1

)eijθ.

(ii) Show that Kn(θ) ≥ 0.(iii) Show that for any continuous 2π periodic function f on R one has

Kn ? f(θ) :=1

2π

∫ π

−πKn(θ − α)f(α)dα

=j=+n∑j=−n

(1− |j|

n+ 1

)(1

2π

∫ π

−πe−ijαf(α)dα

)eijθ.

Sums and Products 9

(iv) Show that Kn ? f(θ)→ f(θ) uniformely in θ as n→∞.

Problem 38. Let λ > 0.(i) Calculate

S1(λ) =∞∑j=0

je−λλj

j!.

(ii) Calculate

S2(λ) =∞∑j=0

j(j − 1)e−λλj

j!.

Problem 39. (i) Let α > 0. Show that

∞∑k=1

(−1)k cos(kx)k2 + α2

=π

2α2· cosh(αx)

sinh(αx)− 1

2α2

where −π ≤ x ≤ π.(ii) Let α > 0. Show that

∞∑k=1

(−1)k sin(kx)k2 + α2

= − π

2α2· sinh(αx)

sinh(απ)

for −π < x < π.

Problem 40. Let a > 0 and b > 0. Show that the continued fraction

a+1b+

1a+

1b+

1a+· · ·

can be written asa

2+

√a2

4+a

b.

Problem 41. Let M,N ≥ 1. Find the sum

M∑m=1

N∑n=1

(m+ n)!(2m+ n)!(m+ 2n)!

.

Problem 42. Study the sum∞∑

`,m,n=1

`

(`m+ `n+mn+ 1)2x`+m+n.


Problem 43. Show that∞∑

n=−∞

1n2 + 3

4n+ 18

= 4π.

Problem 44. Let n ≥ 1. Find the sum

n∑j=1

3j(j + 3)

.

What happens for n→∞?

Problem 45. Let F1 = 1, F2 = 2, Fk+1 = Fk−1 + Fk be the Fibonaccinumbers. Calculate

∞∑k=1

log2(Fk+1)2k+1

.

Problem 46. Let Z be the set of integers. Simplify the sum∑k∈Z

sin(π(x− k)) sin(π(y − k))π(x− k)π(y − k)

.

Problem 47. Let n ≥ 1. One has

(1 + x)n =n∑k=0

(n

k

)xk. (1)

Show that

1 ·(n

1

)+2 ·

(n

2

)+3 ·

(n

3

)+ · · ·+(n−1) ·

(n

n− 1

)+n ·

(n

n

)= n2n−1. (2)

Problem 48. Let n > 2 and A, h > 0. Find Vn given by

Vn =Ah

n3

n−1∑j=1

j2.

Then find limn→∞ Vn. This concern the volume of a pyramide, h is theheight and A is the area of the base.

Sums and Products 11

Problem 49. Let n ≥ 1 and c1, . . . , cn be constants. Find the minimaof the function f : R→ R

f(x) =n∑j=1

(x− cj)2.

Problem 50. Let x ∈ R. The sequence of functions fk(x) is definedby f1(x) = cos(x/2) and for k > 1 by

fk(x) = fk−1(x) cos(x/2k).

Thusfk(x) = cos(x/2) cos(x/22) · · · cos(x/2k).

Obviously, we have fk(0) = 1 for every k. Calculate limk→∞ fk(x) as afunction of x for x 6= 0.

Programming Problems

Problem 1. Let m,n ≥ 2. Calculate the finite sum

Sn,m =m∑j=1

n∑k=1

|j − k|.

Give a C++ implementation.

Problem 2. Let n ∈ N. Calculate the finite sum

Sn =n∑

j=−n

n∑k=−n

|j + k|.

Problem 3. Let m ≥ 1 and n ≥ 1. Find the sum

min(m,n)∑k=0

(n+m− 2k + 1).


1.2 Supplementary Problems

Problem 1. Show that

1 + 2 tanh2(λ) + 3 tanh4(λ) + · · ·+ (n+ 1) tanh2n(λ) + · · · ≡ cosh4(λ).

Note that tanh(0) = 0 and cosh(0) = 1.

Problem 2. Show that∞∑j=2

1j(2j − 1)

= 2 ln(2)− 1.

Problem 3. (i) Show that

14

=1

1 · 2 · 3+

12 · 3 · 4

+1

3 · 4 · 5+ · · ·

(ii) Let n ≥ 1. Show that

11 · 2

+1

2 · 3+ · · ·+ 1

n(n+ 1)=

n

n+ 1.

Problem 4. Let n,m ∈ N and a, b, c, d ∈ R. Show that

(a+ b)n(c+ d)m =n∑r=0

m∑s=0

(n

r

)(m

s

)arbn−rcsdm−s.

Problem 5. Let ` ≥ 1. Study the Gauss sum

G(k, `) =1√`

`−1∑r=0

e2πikr2/`.

Problem 6. Show that if |x| < 1 we have the expansion

1(1− x)2

= 1 + 2x+ 3x2 + 4x3 + · · · .

Let a > 0. Use this expansion to show that

1(a+ x)2

=1a2

(1− 2x

a+

3x2

a2− 4x3

a3+ · · ·

).


Problem 7. Apply mathematical induction to show that

1 + 3 + 5 + · · ·+ (2n− 1) = n2

12 + 42 + 72 + · · ·+ (3n− 2)2 =12n(6n2 − 3n− 1)

13 + 33 + 53 + · · ·+ (2n− 1)3 = n2(2n2 − 1).


ln(2) =1

1 · 2+

13 · 4

+1

5 · 6+ · · ·

Problem 9. Let x > 0. Show that a complicated way to calculate 1/x isgiven by

1x+ 1

+1!

(x+ 1)(x+ 2)+

2!(x+ 1)(x+ 2)(x+ 3)

+ · · ·

Problem 10. Let y ∈ R and byc be the integer part of y. Let ω =(1 +

√5)/2 be the golden mean number (which is an irrational number).

We definexj := j + ((j + 1)/ω)(λ− 1), j = 0, 1, . . .

where λ > 0 and

µk = xk − xk−1, k = 1, 2, . . .

Show that the sequence µ1, µ2, . . . is non-periodic.

Problem 11. Let s > 0 and

f(s) =∫ ∞t=0

ln(1 + st)1 + t2

dt.

Show that

f(s) = (1− ln(s))s+πs2

4+(

ln(s)3− 1

9

)s3 +O(s4).

Problem 12. Let n be an integer and n > 1. Show that

sn = 1 +12

+13

+ · · ·+ 1n


cannot be integer for all n > 1.

Problem 13. Let x ∈ R. Show that the series

x

1 + x2+

x2

1 + x4+

x3

1 + x6+ · · ·

converges absolutely for all values of x, except for +1 and −1.

Problem 14. Show that the series∞∑j=0

zj =1

1− z,

∞∑j=0

jzj =z

(1− z)2,

∞∑j=0

j2zj =z(1 + z)(1− z)3

converge for all |z| < 1.

Problem 15. Let n be an integer with n ≥ 2. Find∑1≤i<j≤n

∣∣∣∣ 12i− 1

3j

∣∣∣∣ .Problem 16. Let ` ≥ 1. Show that∑

j=1

14j2 − 1

=`

2`+ 1

and ∑j=2

1j(j2 − 1)

=14

(1− 2

`(`+ 1)

).

Problem 17. Let 0 ≤ r < 1. Calculate the sum∞∑j=1

rj cos(jφ)

utilizing the identity

cos(jφ) ≡ 12

(eijφ + eijφ).

Problem 18. Calculate the sum∞∑j=1

1j2 + 1


with the knowledge that∞∑j=1

1j2 + 1

≡∞∑j=1

1j2−∞∑j=1

1j2(j2 + 1)

,

∞∑j=1

1j2

=π2

6.


cos(π/4) =12

√2, cos(π/8) =

12

√2 +√

2, cos(π/16) =12

√2 +

√2 +√

2

sin(π/4) =12

√2, sin(π/8) =

12

√2−√

2, sin(π/16) =12

√2−

√2 +√

2.

Problem 20. Let n ≥ 0 and x 6= 0. Find

Sn(x) =n∑k=0

kxk.

Utilize that

Sn+1(x) = Sn(x) + (n+ 1)xn+1 =n∑k=0

(k + 1)xk+1 =n∑k=0

kxk+1 +n∑k=0

xk+1

= xSn(x) +n∑k=0

xk+1

and a case study with x = 1 and x 6= 1.

Problem 21. The Fibonacci numbers are given by xt+2 = xt+1 +xt witht = 0, 1, . . . and x0 = x1 = 1. Consider

F =1x0

+1x1

+1x2

+1x3

+1x4

+1x5

+ · · · ≡ 11

+11

+12

+13

+15

+18

+ · · · .

Show that the series converges. Is F < 4?

Problem 22. Calculate the sum

S =∞∑j=1

2n+ 1n2(n+ 1)

.

Problem 23. Show that the Cantor series

c1 +c22!

+c33!

+c44!

+ · · ·


with 0 ≤ cn ≤ n− 1 is convergent.

Problem 24. Let N ≥ 1 be an integer and k = −4N,−4N + 1, . . . , 4N .We define

f(k) =1N

N∑j=1

cos((

j − 12

)kπ

N

).

Show that f(0) = 1, f(±2N) = −1, f(±4N) = 1 and 0 otherwise.

Problem 25. Show that∑k∈Z

1(x− k)

= π cot(πx).

Note that cot(0) =∞ and cot(π/2) = 0.

Problem 26. Let c > 0. Apply the Poisson summation formula

+∞∑n=−∞

f(n+ a) =+∞∑

k=−∞

exp(2πika)∫ +∞

−∞f(x) exp(−2πikx)dx

for the function

f(x) :=x exp(−cx2) for x ≥ 0

0 for x < 0

Problem 27. Let k,m = 1, 2, . . .. Write a C++ program that implementsthe sum

E(k,m) =1

2k+m−1

m−1∑i=0

(k +m− 1

i

)for a given k and m. The sum plays a role for the arithmetic triangle.

Problem 28. Let L ∈ N0 and κ be a nonnegative real number.(i) Show that

Z(κ) =

(L∑n=0

(L

n

)e−κLn

)2

= (1 + e−κL)2L.

(ii) Show that for κ > 0 one has for large L that Z ≈ 1 + 2Le−κL. Showthat at κ = 0 one has Z(κ = 0) = 22L.


Problem 29. Let n > 2. Show thatn∑j=2

1j≈ ln(n).

Problem 30. Let n ≥ 1 and ω be the primitive nth root of 1. Show thatn−1∑j=0

ωj = 0.


ex + e−x + 2 cos(x) = 4∞∑n=0

x4n

(4n)!.

Problem 32. Show that ∞∑j=0

1j!

( ∞∑k=0

(−1)k

k!

)= 1

Problem 33. Let n ∈ N. Applying the principle of mathematical induc-tion show that

A(n) =n∑k=0

k(k + 1) =n(n+ 1)(n+ 2)

3.

Note that A(0) = 0.

Problem 34. Let N ≥ 2. Find the sumN∑

j1,j2,j3=1j3≥j2≥j1

(j1j2j3 + (j1 + j2 + j3)).

Problem 35. Let h, r > 0 and fixed and m,n ∈ N. Consider

Sm,n =mnr sin(π/n)

√(h

m

)2

+ (r − r cos(π/n))2

= πrsin(π/n)π/n

√h2 +

14π4r2

m2

n4

(sin(π/n)π/n

)4

.


(i) Consider limm,n→∞ Sm,n with m = n2.(ii) Consider limm,n→∞ Sm,n with m = n3.(iii) Consider limm,n→∞ Sm,n with limm→∞,n→∞(m2/n4) = 0 and showthat limm,n→∞ Sm,n = πrh which is a surface area of a half cylinder withradius r and length h.

Problem 36. (i) Let a > 0. Show that

∞∑k=1

cos(kx)k2 + a2

=π

2acosh(a(π − x))

sinh(aπ)− 1

2a.

(ii) Let a > 0. Show that

∞∑k=−∞

1(2k + 1)2 + a2

=π

2atanh

(aπ2

).

Problem 37. Let k ≥ 2. Consider

f(k) = k!(

12!− 1

3!+

14!− · · ·+ (−1)k

k!

)≡ k!

k∑j=2

(−1)j

j!.

Thus f(2) = 1. Find f(3), f(4), f(5), f(6), ... . Which of these numbersis a prime number?

Problem 38. Let M be an integer with M ≥ 2 and =(φ) < 0. Show that

S(k) =M−1∑m=1

e(2πimk)/M

sin2(π(m+ φ)/M)= −4kMe−2πikφ/M

1− e−2πiφ+M2e−2πikφ/M

sin2(πφ).

Note that1

sin2(π(m+ φ)/M)= −4

∞∑p=0

pe−2πi(m+φ)p/M .

Problem 39. The q-exponential function is defined by

exq :=∞∑j=0

xn

[j]!

where

[j] :=qj − q−j

q − q−1.


Find exq for x = 1 and q = 1/2.

Problem 40. Let a, b, c ∈ R. Factorize

(b− c)3 + (c− a)3 + (a− b)3.

The following Maxima may be helpful

T: (b-c)^3 + (c-a)^3 + (a-b)^3;T: expand(T);T: ratsimp(T);

Problem 41. Let n ≥ 0. Starting from

n∑j=0

cjxj = (1 + x)n

show that

c0 + c1 + · · ·+ cn = 2n

c0 − c1 + c2 − c3 + · · ·+ (−1)ncn = 0c1 + 2c2 + 3c3 + · · ·ncn = n2n−1

c20 + c21 + · · ·+ c2n =(2n)!(n!)2

.

Problem 42. Let n ≥ 1. Show that

n∑j=1

ln(1 + 1/j) = ln(n+ 1).

Problem 43. Let x ∈ (−1, 1). Show that

∞∑j=0

jxj =x

1− x)2.

Problem 44. (i) Assuming that

1− 12

+13− 1

4+

15− 1

6+

17− 1

8+ · · · = ln(2).


Show that

1− 12− 1

4+

13− 1

6− 1

8+

15− 1

10− 1

12+ · · · = 1

2ln(2).

Note that the convergence of the first series is not absolute.(ii) Show that

1 +13

+15

+ · · ·+ 12n+ 1

− 12

ln(n)

tends to a finite value as n→∞. Find this value.

Problem 45. Let β > 0. Show that

∞∑n=0

e−βn =1

1− e−β.

Problem 46. Let n be a positive integer and f(θ1, . . . , θN ) be a periodicfunction, i.e. periodic 2π for each θj (j = 1, . . . , N). Show for large n wehave (

N∏k=1

∫ 2π

0

dθk

) N∑j=1

cos(θj)

n

f(θ1, . . . , θN )

≈ Nn

(N∏k=1

∫ ∞−∞

dθke−n(θk)2/(2N)

)f(θ1, . . . , θN ).

Problem 47. Let A, B be finite sets and n(A), n(B) the numbers ofelements in A and B, respectively. Is

n(A) + n(B) = n(A ∪B) + n(A ∩B) ?

Prove or disprove.

Problem 48. Let f : [0, 1] → [0, 1] be a differentiable function, forexample the logistic map f(x) = 4x(1− x). Let f (k) be the k-th iterate ofthe function f . Let

bk := sup0≤x≤1

∣∣∣∣ ddxf (k)(x)∣∣∣∣ .

Show thatlimk→∞

(bk)1/k

exist. Apply the chain rule.



n∑k=1

k

(n

k

)= n2n−1.

Problem 50. Is the series

S =∞∑k=1

k

1 + 2k3

convergent?

Problem 51. Give a non-trival infinite sequence (x0, x1, x2, . . .) such that

∞∑j=0

|xj |1 + |xj |

is finite.


∞∑j=−∞

(−1)j1

j2 + a2=

π

a sinh(πa).

Problem 53. Let n = 0, 1, 2, . . .. The Fermat numbers are given by

Fn = 2(2n) + 1.

Show that the Fermat numbers satisfy the recurrence relation

Fn+1 = (Fn − 1)2 + 1

with F0 = 3

Problem 54. Let N ≥ 1. Consider the Hilbert space L2([−1, 1]). TheChebyhev-Gauss-Lobatto points are given by

xj = cos(jπ/N), j = 0, 1, . . . , N.

Find the four points for N = 3.


Problem 55. Let |z| < 1. Show that∞∑j=1

jzj = (1− z)−1∞∑j=1

zj .

Show that∞∑j=1

j2zj = (1 + z)(1− z)−2∞∑j=1

zj


13

+29

+ · · ·+ 2n

3n+1+ · · · = 1

3

∞∑n=0

(23

)n=

13

11− 2/3

= 1.

Problem 57. (i) Let a > 0. Show that∞∑n=0

1(a+ n)(a+ n+ 1)

=1a.

(ii) Show that∞∑n=1

1n(n+ 1)(n+ 2)

=14.

(iii) Let n ≥ 2. Show that

n−1∑j=0

1(j + 1)(j + 2)

= 1− 1n+ 1

and thus show that∞∑j=0

1(j + 1)(j + 2)

= 1.

Problem 58. Let n be a positive integer. Show that

sin(2nα)2n sin(α)

= cos(α) cos(2α) cos(22α) · · · cos(2n−1α).

Problem 59. Let Γ(x) be the gamma function. Show thatn∑k=0

(−1)k(n

k

)(α+ 2k)Γ(α+ k)Γ(α+ k + n+ 1)

= δn,0.


Problem 60. Let a > 0 and a /∈ N. Show that

cos(ax) =2a sin(ax)

π

(1

2a2− cos(x)a2 − 12

+cos(2x)a2 − 22

− cos(3x)a2 − 32

+ · · ·)

for all x ∈ [−π, π].

Problem 61. (i) Let n ∈ N. Show that

14 + 24 + · · ·+ n4 =130n(n+ 1)(2n+ 1)(3n2 + 3n− 1).

(ii) Let n ≥ 1. Show that

n∑k=0

k4 =n5

5+n4

2+n3

3− n

30.

Problem 62. Let N ≥ 1. Find the sums

vk =N−1∑j=0

(exp

(2πNjk

)− 1), k = 0, 1, . . . , N − 1.

Problem 63. (i) Let ` be a positive integer and φ ∈ R. Show that

+∑m=−`

eimφ =sin((`+ 1/2)φ)

sin(φ/2).

(ii) The function

Dn(θ) :=sin((n+ 1

2 )θ)sin( 1

2θ)

is called the Dirichlet kernel. Show that

Dn(θ) =k=+n∑k=−n

eikx.

Problem 64. Let k be a positive integer and k ≥ 2. Show that

1 + x− 2xk ≡ (1− x) + 2x(1− xk−1) ≡ (1− x)

1 + 2k−1∑j=1

xj

.


Problem 65. Let n ≥ 2. Let Sn be the standard n-simplex embedded inRn

Sn :=

x ∈ Rn :n∑j=1

xj = 1, for xk ≥ 0, for k = 1, . . . , n

.

We denote by Sk,n−1 the kth face of the boundary of Sn. They are (n−1)-simplexes

Sk,n−1 := x ∈ Rn : xk = 0, x ∈ Sn for k = 1, . . . , n.

Show that the boundary ∂Sn of Sn is the union

∂Sn = ∪nj=1Sj,n−1

of the faces.

Problem 66. Show that∞∏j=1

cos( x

2j)

=sin(x)x

for x ∈ R.

Problem 67. The triple sum

∞∑k1=−∞

∞∑k2=−∞

∞∑k3=−∞

′ (−1)k1+k2+k3√k2

1 + k22 + k2

3

plays a role in solid state physics (Madelung constant). Here ′ indicatesthat the term (k1, k2, k3) = (0, 0, 0) is omitted. Given a positive integer N .Write a C++ program that implements the sum

N∑k1=−N

N∑k2=−N

N∑k3=−N

′ (−1)k1+k2+k3√k2

1 + k22 + k2

3

.

Run the program for different N and compare with the exact value.

Problem 68. Find the sum

SL =L∑n=1

exp(iπ

2Ln2

)for L = 1, 2, 3.


Problem 69. Show that the series∞∑j=1

sin(jx)√j

converges for all x ∈ R.

Problem 70. Let k ≥ 0 and s > 0. Show that

k∏`=0

s+ `+ 1s+ `

=s+ k + 1

s.

Problem 71. Let n be a positive integer. Give a C++ implementationof the sum

n∑k=0

n∑`=0

(k

`

)using templates so that the Verylong class of SymbolicC++ can be used.

Problem 72. Let n,m ≥ 1. Implement the sum

Sn,m :=(2m)!22mm!

m∑k=0

2k(n

k

)(m

k

)using the Verylong class and Rational class of SymbolicC++.

Problem 73. Let n ∈ N and m1,m2, . . . ,mn ∈ N0. Consider the function

f(m1,m2, . . . ,mn) :=δm1,0 for n = 1

δm2,0δm1,1 for n = 2δmn,0δ(m1+···+mn),n−1

∏n−2j=1 H(j −

∑jk=1mn−k) for n ≥ 3

where H is the step function

H(x) :=

1 for x ≥ 00 for x < 0

Give an implementation of this function in SymbolicC++ for a given nusing the Verylong class.

Chapter 2

Maps

2.1 Solved ProblemsProblem 1. Newton’s sequence takes the form of a difference equation

xt+1 = xt −f(xt)f ′(xt)

where t = 0, 1, 2, . . . and x0 is the initial value at t = 0. Let f : R → R begiven by

f(x) = x2 − 1

and x0 6= 0. Find the fixed points of f . Find the fixed points of thedifference equation. Let x0 = 1/2. Find x1, x2, x3. Find Newton’s sequencefor this function. Obtain the exact solution of the difference equation.

Problem 2. (i) Solve the nonlinear recurrence relation

xn+1 = x2n, n = 0, 1, . . .

where x0 = 2.(ii) Solve the linear recurrence relation

xn+1 = xn + xn−1 + xn−2, n = 2, 3, . . .

and the initial values x0 = x1 = x2 = 1.

26

Maps 27

(iii) Solve the linear recurrence relation

xn+1 = 1 +n−1∑j=0

xj , x0 = 1.

Problem 3. Let x > 0 and p > 0. Consider the map

f(x) = xep−x.

(i) Find the fixed points. Study the stability of the fixed points.(ii) Show that f has a least one periodic point x∗ with x∗ 6= 0 or p.

Problem 4. Let f : R → R+ be a positive, continously differentiablefunction, defined for all real numbers and whose derivative is always neg-ative. Show that for any real number x0 (initial value) the sequence (xk)obtained by Newton’s method

xk+1 = xk −f(xk)f ′(xk)

has always limit ∞.

Problem 5. Let f : R→ R be a continuosly differentiable map. Let f (n)

be the n-th iterate of f . Calculate the derivative of f (n) at x0.

Problem 6. (i) Solve the second order linear difference equation

xt+2 = xt+1 + xt t = 0, 1, 2, . . .

where x0 = 0 and x1 = 1.(ii) Give the definition of the golden mean number and derive this number.(iii) Calculate

limt→∞

xt+1

xt.

Problem 7. The recursion relation

Fn+2 = Fn + Fn+1

with the initial values F0 = F1 = 1 provides the Fibonacci sequence1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .. A generalization of the Fibonacci sequenceis the q-analogue of the sequence defined by the recursion relation

Fn+2(q) = Fn(q) + qFn+1(q)


and the initial condition F0(q) = 1 and F1(q) = q. Here, q is a real orcomplex number.(i) Give the first five terms of the sequence.(ii) Find a generating function of Fn(q).(iii) Find an explicit expression for Fn(q).

Problem 8. (i) Find the linear map f : 0, 1 → −1, 1 such that

f(0) = −1, f(1) = 1. (1)

(ii) Find a linear map g : −1, 1 → 0, 1 such that

g(−1) = 0, g(1) = 1. (2)

This is obviously the inverse map of f .

Problem 9. Consider the differentiable function f : R→ R

f(x) =1− cos(2πx)

x

where using L’Hospital f(0) = 0.(i) Find the zeros of f .(ii) Find the maxima and minima of f .

Problem 10. Given two manifolds M and N , a bijective map φ from Mto N is called a diffeomorphism if both φ : M → N and its inverse φ−1 aredifferentiable. Let f : R→ R be given by the analytic function

f(x) = 4x(1− x)

and the analytic function g : R→ R be given by

g(x) = 1− 2x2.

(i) Can one find a diffeomorphism φ : R→ R such that

g = φ f φ−1 ?

(ii) Consider the diffeomorphism ψ : R→ R

ψ(x) = sinh(x).

Calculate ψ g ψ−1.

Maps 29

Problem 11. Consider the polynomial p(x) = x3−3x+3. Show that forany positive integer N , there is an initial value x0 such that the sequencex0, x1, x2, . . . obtained from Newton’s method

xn+1 = xn −p(xn)p′(xn)

=2x3

n − 33(x2

n − 1), n = 0, 1, 2, . . .

has period N .

Problem 12. Consider the linear recursion equation for F (n) with n =0, 1, 2, . . .

(n+ 1)F (n+ 1)− nF (n− 1) + F (n) = 0

where F (0) = 0, F (1) = 1. We define

f(z) :=∞∑n=1

F (n)zn

where z is an undeterminate. Find the differential equation for f with theinitial condition. Solve the differential equation.

Problem 13. Let f : Rn → Rn be an analytic function. Consider themap

xj = f(xj−1) = · · · = f(x0).

To study the evolution of phase-space distributions, we can introduce theevolution operator U(x′,x, j) such that any initial phase-space distributionρ(x, 0) evolves into

ρ(x′; j) =∫

Ω

U(x′,x; j)ρ(x, 0)dx

where Ω is the phase space area. Find U(x′,x; j).

Problem 14. Find the solution of the recursion relation

aj+2 =23aj+1 +

13aj , j = 1, 2, . . .

with a1 = 5, a2 = 1. Calculate limj→∞ aj .

Problem 15. Solve the linear difference equation

xt+1 = 2xt − t, t = 0, 1, 2, . . .

with the initial value x0 = 1.


Problem 16. LetN0 := 0, 1, 2, . . .

Find an invertible map f : N0 × N0 × N0 → N0 with

f(0, 0, 0) = 0, f(0, 0, 1) = 1, f(0, 1, 0) = 2, f(0, 1, 1) = 3,

f(1, 0, 0) = 4, f(1, 0, 1) = 5, f(1, 1, 0) = 6, f(1, 1, 1) = 7

etc. Find the inverse function.

Problem 17. Consider the second order difference equation

xt+2 = xt+1xt

with the initial values x0 = a, x1 = b. Give the solution. What are thefixed points?

Problem 18. Let A be a set. Suppose that it is possible to define subsetsA1, A2, . . . of A which have the properties that(i) the sets are pairwise disjoint; that is Ai ∩ Aj = ∅ for all i, j = 1, 2, . . .and j 6= i

(ii) A1 ∪A2 ∪ · · · = A.

Then the family of sets A1, . . . is called a partition of A.

Let S = 1, 2, 3, . . . , 9 and A = 1, 4, 7. B = 2, 3, 5, 6, C = 7, 8, 9. IsA,B,C a partition of S?

Problem 19. Let S be a finite set. Let P(S) be the power set of S. ThenP(S) has 2|S| elements.(i) Show that the composition

A B := (A ∪B) ∩ (A ∪ B)

defines a group, where A,B ∈ P(S). Here denotes the complement.(ii) Show that the composition

A B := (A ∩B) ∪ (A ∩ B)

defines a group, where A,B ∈ P(S). Here denotes the complement.

Problem 20. Find a function f : [0, 1]→ (0, 1] that is one-to-one.

Maps 31

Problem 21. Let n ∈ N. Consider the map

ut+1 =12

(ut +

n

ut

), t = 0, 1, 2, . . .

given the initial value u0 with u0 > 0. Show that

ut+1 −√n

ut+1 +√n

=(ut −

√n

ut +√n

)2

.

Show that ut →√n as t→∞. Show that

√n is a fixed point.

Problem 22. Let 0 ≤ α < π/4. Consider the transformation

X(x, y, α) =1√

cos(2α)(x cos(α) + iy sin(α))

Y (x, y, α) =1√

cos(2α)(−ix sin(α) + y cos(α)).

(i) Show that X2 + Y 2 = x2 + y2.(ii) Do the matrices

1√cos(2α)

(cos(α) i sin(α)−i sin(α) cos(α)

)form a group under matrix multiplication?

Problem 23. A smooth function f : Rn → R is called homogeneous ofdegree r if

f(εx1, . . . , εxn) = εrf(x1, . . . , xn). (1)

Show that (Euler’s identity)

n∑j=1

∂f

∂xjxj = rf.

Problem 24. Find the invariance group of the function f : R→ R

f(x) = sin(x).

Problem 25. Consider the analytic function f : R→ R

f(x) = cos(sin(x))− sin(cos(x)).


Show that f admits at least one critical point. By calculating the secondorder derivative find out whether this critical point refers to a maxima orminima.

Problem 26. Let f : R2 → R, g : R2 → R be analytic functions. Wedefine the star product

f(x1, x2) ? g(x1, x2) :=

limx′1→x1,x′2→x2

exp(

∂

∂x1

∂

∂x′2− ∂

∂x′1

∂

∂x2

)f(x1, x2)g(x′1, x

′2).

Letf(x1, x2) = sin(x1 + x2), g(x1, x2) = sin(x1 − x2).

Find the star product.

Problem 27. LetN1, N2 be given positive integers. Let n1 = 0, 1, . . . , N1−1, n2 = 0, 1, . . . , N2 − 1. There are N1 ·N2 points. The points (n1, n2) area subset of N0 × N0 and can be mapped one-to-one onto a subset of N0

j(n1, n2) = n1N2 + n2

where j = 0, 1, . . . , (N1 − 1)(N2 − 1). Find the inverse of this map.

Problem 28. Let a ∈ R. Consider the transformation

t(t, x) =1aeax sinh(at), x(t, x) =

1a

(eax cosh(at)− 1)

with lima→0 t = t, lima→0 x = x. Find the inverse of the transformation.

Problem 29. Show that the map f : (0, 1)→ R

x 7→ f(x) =x− 1/2x(x− 1)

is bijective. Note that f(1/2) = 0 and f(1/4) = 4/3.

Problem 30. Consider the map

f : R→ (x1, x2) : x21 + (x2 − 1)2 = 1 ∧ x2 6= 2

defined by

f(t) =(

4tt2 + 4

,2t2

t2 + 4

).

Maps 33

Show that f is bijective. Show that f and f−1 are continuous.

Problem 31. Let N ≥ 2 and δ := 1/(N + 1). Solve the linear one-dimensional linear difference equation

xj+1 − 2xj + xj−1 = −12δ4(j + 1)2, j = 0, 1, . . . , N − 1

with the boundary conditions x−1 = xN = 0.

Problem 32. (i) Consider the analytic map f : R→ R

f(x) = x(1− x).

Find the fixed points of f . Are the fixed points stable? Prove or disprove.(ii) Let x0 = 1/2 and iterate

f(x0), f(f(x0)), f(f(f(x0))), . . .

Does this sequence tend to a fixed point? Prove or disprove.(iii) Let x0 = 2 and iterate

f(x0), f(f(x0)), f(f(f(x0))), . . .

Does this sequence tend to a fixed point? Prove or disprove.(iv) Find the critical points of f . Then find the extrema of f .(v) Find the roots of f , i.e. solve f(x) = 0.(vi) Find the minima of the function g(x) := |f(x)|.

Problem 33. Let c > 0 and 0 ≤ v1 < c, 0 ≤ v2 < c. We define thecomposition

v1 ? v2 :=v1 + v2

1 + v1v2/c2.

Is the composition associative?

Problem 34. Solve the initial value problem of the system of first orderdifference equations

x1,t+1 =−2x1,t

x2,t+1 =89x1,t − x2,t.

The first difference equation is independent of x2,t and we find the solutionof the initial value problem

x1,t+1 = (−2)tx1,0.


Problem 35. Let mA, mB , RA, RB be the masses and centre-of masscoordinates of mass A and B, respectively. We set m = mA + mB . Findthe inverse of the transformation

r(RA,RB) = RA −RB , R(RA,RB) =1m

(mARA +mBRB).

Problem 36. Consider the map f : N0 × N0 → N0 × N0

f1(n1, n2) = |n1 − n2|, f2(n1, n2) = n1 + n2.

Is the map invertible?

Problem 37. Let n,m ≥ 0. Consider the differential operators

Kn :=n∑j=0

uj(x)dj

dxj, Lm :=

m∑j=0

vj(x)dj

dxj

where the uj(x)’s and vj(x)’s are smooth functions. If the two differentialoperators Kn and Lm commute, then there is a nonzero polynomial R(z, w)such that R(Kn, Lm) = 0. The curve Γ defined by R(z, w) = 0 is called thespectral curve. If we consider the eigenvalue problem

Knψ = zψ, Lmψ = wψ

then (z, w) ∈ Γ.(i) Let (α ∈ R)

K =(d2

dx2+ x3 + α

)2

+ 2x,

L =(d2

dx2+ x3 + α

)3

+ 3xd2

dx2+ 3

d

dx+ 3x(x2 + α).

Show that K and L commute with w2 = z3 − α.(ii) Show that

K =(d3

dx3+ x2 + α

)2

+ 2d

dx

and

L =(d3

dx3+ x2 + α

)3

+ 3d4

dx4+ 3(x2 + α)

d

dx+ 3x

commute.

Problem 38. Let A, B, C be nonempty sets. Let f : A → B andg : B → C are invertible maps. Then the composition of the functions

Maps 35

g f : A→ C in invertible and (g f)−1 = f−1 g−1. Function compositionis associative.

Problem 39. Consider the map f : (−1, 1)→ R

f(x) =x

1− |x|.

Show that the map f−1 : R→ (−1, 1) is given by

f−1(x) =x

1 + |x|.

Problem 40. Let τ ∈ R. Show that the curves

x(τ) =(x1(τ)x2(τ)

)=(

1τ

), y(τ) =

(y1(τ)y2(τ)

)=(ττ2

)are linearly independent.

Problem 41. Let r, k be positive integers with r ≤ k and M =x1, . . . , xr and N = y1, . . . , yk. Then there

k!(k − r)!

surjective maps f : M → N . Find all the maps for r = k = 2.

Problem 42. Consider the first order difference equation

xτ+1xτ + 2xτ + xτ+1 = 0, τ = 0, 1, 2, . . .

with x0 6= 0. Show that this difference equation can be linearized withxτ = 1/yτ where x0 6= 0. First show that x∗ = 0 and x∗ = −3 are solutionsof the nonlinear difference equation.


Problem 1. Let x,y ∈ R3 and × be the vector product

x× y =

x2y3 − x3y2

x3y1 − x1y3

x1y2 − x2y1

.


Hence the vector product is a map from R3 × R3 into R3. Show that themap is differentiable.

Problem 2. Study the map (Legendre map) f : R2 → R2

f1(x1, x2) = ex1 cos(x2), f2(x1, x2) = −ex1 sin(x2).

Find the functional determinant.

Problem 3. Consider the four symbols A,B,C,D. The Rudin-Shapirosubstitution is

A 7→ AC, B 7→ DC, C 7→ AB, D 7→ DB.

Find the sequence starting of with B.

Problem 4. Solve the difference equation

xt+1 = txt + t2 mod 2

with t = 0, 1, 2, . . . and x0 = 1.

Problem 5. Study the map f : Z× Z× Z→ Z× Z× Z

f1(x1, x2, x3) = x1 − x2x3, f2(x1, x2, x3) = −x2 + x1x3,

f3(x1, x2, x3) = x3 − x1x2.

(i) Show that the fixed points are

(0, 0, 0), (1, 0, 0), (0, 0, 1), (−1, 0, 0), (0, 0,−1).

(ii) Show that (x1 = 2, x2 = 2, x3 = 2) provides a periodic orbit.(iii) Show that (x1 = 1, x2 = 2, x3 = 1) provides an eventually periodicorbit.(iv) Is the map invertible?

Problem 6. Consider the map f : [0, 1]3 → [0, 1]3 given by

f1(x1, x2, x3) = 2x1, f2(x1, x2, x3) =12x2, f3(x1, x2, x3) =

12x3

for 0 ≤ x1 ≤ 1/2 and

f1(x1, x2, x3) = 2x1−1, f2(x1, x2, x3) =12

(x2+1), f3(x1, x2, x3) =12

(x3+1)

Maps 37

for 1/2 < le ≤ 1. Find the orbit for x1 = x2 = x3 = 1/3.

Problem 7. Consider the map f : R2 → R2

f1(x1, x2) = sinh(x1) sin(x2), f2(x1, x2) = cosh(x1) cos(x2).

Find the fixed points of the map and study their stability.

Problem 8. Study the system of difference equations

x0,t+1 = 2x0,t − x1,t − x2,t − x3,t − x4,t

x1,t+1 =−12x1,t + x0,t

x2,t+1 =−12x2,t + x0,t

x3,t+1 =−12x3,t + x0,t

x4,t+1 =−12x4,t + x0,t

with the initial conditions

x0,0 = 1, x1,0 = x2,0 = x3,0 = x4,0 = 0.

Problem 9. Consider the difference equation

xt+1 = 2xt − 2, t = 0, 1, 2, . . .

where x0 = 1. Find the general solution. Show that x∗ = 2 is a particularsolution (the fixed point).

Problem 10. The Fibonacci numbers can be defined by(xt+2

xt+1

)=(

1 11 0

)(xt+1

xt

)with x0 = x1 = 1. Find the eigenvalues of the 2 × 2 matrix. Study thesequence

xt+4

xt+3

xt+1

xt

= (A⊗A)

xt+3

xt+2

xt+1

xt

with x0 = x1 = x2 = x3 = 1 and ⊗ denotes the Kronecker product.


Problem 11. Let x0 ∈ R+ and r ∈ R+. Show that

xj+1 =12

(xj +

r

xj

), j = 0, 1, 2, . . .

tends to√r.

Problem 12. Let A = (ajk) be a 3×3 matrix with det(A) 6= 0. Considerthe map f : R2 → R2 given by

f1(x1, x2) =a11x1 + a12x2 + a13

a31x1 + a32x2 + a33, f2(x1, x2) =

a21x1 + a22x2 + a23

a31x1 + a32x2 + a33,

Find the inverse of the map.


f1(x1, x2) = x1 + x2, f2(x1, x2) = x1x2

or written as difference equation

x1,τ+1 = x1,τ + x2,τ , x2,τ+1 = x1,τx2,τ .

Find the fixed points of the map. Then solve the difference equation forx1,0 = 1/4, x2,0 = 1/5.

Problem 14. Show that(∂F (x, y)∂x

)2

+(∂F (x, y)∂y

)2

expressed in polar coordinates x = r cos(θ), y = r sin(θ) takes the form(F (x(r, θ), y(r, θ)

∂r

)2

+1r

(∂F (x(r, θ), y(r, θ))

∂θ

)2

.

Problem 15. Let r > 1 and τ = 0, 1, 2, . . .. Study the system of differenceequation

x1,τ+1 =1rex2,τ , x2,τ+1 = re−x1,τ

with the initial conditions x1,0 = x2,0 = 0. Are there fixed points?

Problem 16. Let m ∈ N. Solve the recurrence relation

f(n) = 1 + f(n/2) for n ≥ 2, n = 2m

Maps 39

with the initial condition f(1) = 1.

Problem 17. Study the initial value problem second order differenceequation

xτ+2 = xτ+1 + xτ − xτ+1xτ

with x0 = 1, x1 = 1/2.

Problem 18. A set A is equilvalent to the set B (one writes A ∼ B) ifthere is a mapping f : A→ B which is (1, 1) and onto. Show that R2 ∼ R.Utilize that

(·a1a2a3 . . . , ·b1b2b3 . . .)↔ ·a1b1a2b2 . . .

defines a (1, 1) map between pairs of decimal expansions and single expres-sions of numbers in the interval (0, 1).

Problem 19. Let Ω = [0, 1), T (x) = 2x (mod 1), and µ the Lebesguemeasure. Show that T is ergodic.

Problem 20. (i) Let c > 0. Study the difference equation

xτ+1 =xτ

c+ xτ, τ = 0, 1, . . .

with x0 > 0. First find the fixed points.(ii) Study the difference equation

xτ+1 =6xτ

(1 + xτ )2

with x0 > 0. First find the fixed points.

Problem 21. Let x ∈ [0, 1] and bac be the integer part of a. Show thatthe sequence xt (t = 0, 1, . . .) given by

x0 = 0, and xt+1 = xt +1

2t+1b2t+1(x− xt)c

converges to x.

Problem 22. The Kustaanheimo-Stiefel transformation is defined by themap from R4 (coordinates u1, u2, u3, u4) to R3 (coordinates x1, x2, x3)

x1(u1, u2, u3, u4) = 2(u1u3 − u2u4)x2(u1, u2, u3, u4) = 2(u1u4 + u2u3)x3(u1, u2, u3, u4) = u2

1 + u22 − u2

3 − u24


together with the constraint

u2du1 − u1du2 − u4du3 + u3du4 = 0.

(i) Show that

r2 = x21 + x2

2 + x23 = u2

1 + u22 + u2

3 + u24.

(ii) Show that

∆3 =14r

∆4 −1

4r2V 2

where

∆3 =∂2

∂x21

+∂2

∂x22

+∂2

∂x23

, ∆4 =∂2

∂u21

+∂2

∂u22

+∂2

∂u23

+∂2

∂u24

and V is the vector field

V = u2∂

∂u1− u1

∂

∂u2− u4

∂

∂u3+ u3

∂

∂u4

(iii) Consider the differential one form

α = u2du1 − u1du2 − u4du3 + u3du4.

Find dα. Find LV α, where LV (.) denotes the Lie derivative.(iv) Let g(x1(u1, u2, u3, u4), x2(u1, u2, u3, u4), x3(u1, u2, u3, u4)) be a smoothfunction. Show that LV g = 0.

Problem 23. The Kustaanheimo-Stiefel map R4 → R3 ((u1, u2, u3, u4)→(x1, x2, x3)) is given by

x1 = 2(u1u3 − u2u4), x2 = 2(u1u4 + u2u3), x3 = u21 + u2

2 − u23 − u2

4

together with the constraint

α ≡ u2du1 − u1du2 − u4du3 + u3du4 = 0.

Show that applying the Kustaanheimo-Stiefel map the Laplacian operator∆3 in R3 can be written as

∆3 =14r

∆4 −1

4r2V 2

where

r = (x21+x2

2+x23)1/2 = u2

1+u22+u2

3+u24, V = u2

∂

∂u1−u1

∂

∂u2−u4

∂

∂u3+u3

∂

∂u4.

Chapter 3

Functions

3.1 Solved ProblemsProblem 1. Let fj : Rn → R, j = 1, 2, . . . , n be real-valued functionswith continuous second-order partial derivatives everywhere on Rn. Sup-pose that there are constants cij such that

∂fi∂xj− ∂fj∂xi

= cij

for all i and j, 1 ≤ i ≤ n. Prove that there is a function g : Rn → Rsuch that fi + ∂g/∂xi is linear for all i = 1, 2, . . . , n. A linear functionp : Rn → R is of the form

p(x) = a0 + a1x+ a2x2 + · · ·+ anxn.

Problem 2. Let f : Rn → R be a differentiable function and x ∈ Rn.Consider the invertible n × n matrix R and the transformation x′ = Rx.The transformation operator PR associated with the invertible matrix R isdefined by

PRf(x) := f(R−1x).

Note that the operator acts upon the coordinates x and not on the argumentof f . This means PRPSf(x) = f(S−1R−1x), where S is another invertiblen× n matrix. Show that

PRPSf(x) = PRSf(x).

41


Problem 3. A ratio list is a finite list of positive numbers, (r1, r2, . . . , rn).An iterated function system realizing a ratio list (r1, r2, . . . , rn) in a metricspace S is a list (f1, f2, . . . , fn), where fj : S → S is a similarity withratio rj . A nonempty compact set K ⊆ S is an invariant set for theiterated function system (f1, f2, . . . , fn) iff K = f1(K)∪f2(K)∪· · ·∪fn(K).The triadic Cantor set is an invariant set for an iterated function systemrealizing the ratio list (1/3, 1/3). The Sierpinski gasket is an invariant setfor an iterated system realizing the ratio list (1/2, 1/2, 1/2). The dimensionassociated with a ratio list (r1, r2, . . . , rn) is the positive number s such thatrs1 + rs2 + · · · + rsn = 1. Let (r1, r2, . . . , rn) be a ratio list. Suppose eachrj < 1. Show that there is a unique nonnegative number s satisfying

n∑j=1

rsj ≡n∑j=1

es ln(rj) = 1.

The number s is 0 iff n = 1.

Problem 4. The arc length of the equilateral hyperbola

h(t) =√t2 − 1, t ≥ 1

starting at t = 1 is given by

Lh(x) =∫ x

1

√2t2 − 1t2 − 1

dt

as a function of the terminal point t = x. The tangent line to the hyperbolaat t = x is

Th(t) =√x2 − 1 +

x√x2 − 1

(t− x)

whose intersection with the t-axis is t = 1/x (t ∈ (0, 1)). The line

Nh(t) = −√x2 − 1x

t

is perpendicular to Th passing through the origin.(i) Find the point Ph where the lines Th and Nh intersect.(ii) Calculate the distance from (x, h(x)) to the common point Ph.

Problem 5. Given a differentiable function f . The logarithmic derivativeof f is defined as

1f

df

dx.

Functions 43

When f is a real function of real variables and takes strictly positive valuesthen the chain rule provides

d

dxln(f(x)) =

1f

df

dx.

The logarithmic derivative has the following properties.1. The logarithmic derivative of the product of functions is the sum of theirlogarithmic derivatives

1fg

d

dx(fg) =

1f

df

dx+

1g

dg

dx

2. The logarithmic derivative of the quotient of functions is the differenceof their logarithmic derivatives

1f/g

d

dx(f/g) =

1f

df

dx− 1g

dg

dx

3. The logarithmic derivative of the α-th power of a function f is α timesthe logarithmic derivative of the function

1fα

d

dx(fα) = α

1f

df

dx

4. The logarithmic derivative of the exponential of a function equals thederivative of a function

1ef

d

dxef =

df

dx

(i) Find the logarithmic derivative of f : R→ R, f(x) = cosh(x).(ii) Let f be a meromorphic function in the open and conncted set D ⊆ C.Let G ⊆ D be a region such that its closure G ⊆ D and its boundary ∂G isa continuous curve not containing a zero or pole of f . Let N be the numberof zeros of f lying inside G and P be the number of poles of f lying insideG. Then

N − P =1

2πi

∫∂G

1f(z)

df(z)dz

dz

where ∂G is an oriented boundary of G. Calculate the left and right-handside for the function f(z) = 1/z and G = (x, y) : x2 + y2 ≤ 1 .

Problem 6. Let f be an analytic function. Let p ∈ N and α ∈ R. Showthat (

d

dx+ αx

)pf ≡ exp

(−αx

2

2

)dp

dxp

(exp

(αx2

2

)f

).


Problem 7. Consider

K(t, s) =

1 if 0 ≤ t ≤ s0 if s ≤ t ≤ 1 .

Show that this kernel satisfies the functional equation

K(cu+ a, t) = K

(u,t− ac

)where c > 0.

Problem 8. Find a continuous differentiable function f : R → R whichhas no fixed points and no critical points.

Problem 9. Let a, b, p, q, r ∈ R, b > a, a < p, p < r, r < q, q < b and

p− a = r − p, q − r = b− q.

In fuzzy logic the following membership function plays an important role

f(x; a, b, r, p, q) =

0 x ≤ a2m−1((x− a)/(r − a))m a < x ≤ p1− 2m−1((r − x)/(r − a))m p < x ≤ r1− 2m−1((x− r)/(b− r))m r < x ≤ q2m−1((b− x)/(b− r))m q < x < b0 x ≥ b

where m is the fuzzifier. Im most cases m = 2. Where are the crossoverpoints? What is the value at the centre?

Problem 10. Let f : R2 → R2 be the mapping of (x, y) → (u, v) givenby

u(x, y) = ex cos(y), v(x, y) = ex sin(y).

What is the range of f? Show that the Jacobian determinant is not zero atany point of R2. Thus every pont of R2 has a neighbourhood in which f isone-to-one. Nevertheless f is not one-to-one on R2. What are the imagesunder f of lines parallel to the coordinate axes?

Problem 11. Let x, y ∈ R. Calculate the square root√x2 + y2 − 2xy.

Functions 45

Problem 12. A criterion for linearly independence of functions f0, f1, . . . , fn ∈Cn[a, b] is that the Wronski determinant is nonzero

det

f0 f1 . . . fnf ′0 f ′1 . . . f ′n...

.... . .

...f

(n)0 f

(n)1 . . . f

(n)n

6= 0.

Here ′ denotes derivative. Apply this criterion to the functions

f0(x) = cos(x), f1(x) = sin(x).

Problem 13. For a sphere of radius r and mass density ρ the mass thatmust be concentrated at its centre is (λ ≥ 0)

M(λ) =4πrρλ2

(cosh(λr)− sinh(λr)/(λr)).

Find limλ→0M(λ).

Problem 14. Let −1 < a < 1. Find the inverse of the transformation

λ(z) =z − a1− az

.

Problem 15. Consider the Jacobi elliptic functions

sn(x, k), cn(x, k), dn(x, k)

where x ∈ R and k2 ∈ [0, 1]. We have

sn(x, 0) = sin(x), cn(x, 0) = cos(x), dn(x, 0) = 1

and

sn(x, 1) =ex − e−x

ex + e−x= tanh(x), cn(x, 1) = dn(x, 1) =

2ex + e−x

≡ sech(x).

(i) Find an expression using Jacobi elliptic functions that interpolates be-tween sin(x) for k = 0 and sinh(x) for k = 1.(ii) Find an expression using Jacobi elliptic functions that interpolates be-tween cos(x) for k = 0 and cosh(x) for k = 1.(iii) Use this result to interpolate between the matrices(

cos(x) sin(x)− sin(x) cos(x)

)and

(cosh(x) sinh(x)sinh(x) cosh(x)

).


Problem 16. Let f1, f2 be differentiable functions and f1(x) > 0, f2(x) >0 for all x. Let f(x) = f1(x)f2(x). Find

d

dx(ln(f(x)).

Problem 17. Let x ∈ R. The sequence of functions fk(x) is definedby f1(x) = cos(x/2) and for k > 1 by

fk(x) = fk−1(x) cos(x/2k).

Thusfk(x) = cos(x/2) cos(x/22) · · · cos(x/2k).

Obviously, we have fk(0) = 1 for every k. Calculate limk→∞ fk(x) as afunction of x for x 6= 0.

Problem 18. (i) Consider the transformation in R3

x0(a, θ1) = cosh(a)x1(a, θ1) = sinh(a) sin(θ1)x2(a, θ1) = sinh(a) cos(θ1)

where a ≥ 0 and 0 ≤ θ1 < 2π. Find

x20 − x2

1 − x22.

(ii) Consider the transformation in R4

x0(a, θ1, θ2) = cosh(a)x1(a, θ1, θ2) = sinh(a) sin(θ2) sin(θ1)x2(a, θ1, θ2) = sinh(a) sin(θ2) cos(θ1)x3(a, θ1, θ2) = sinh(a) cos(θ2)

where a ≥ 0, 0 ≤ θ1 < 2π and 0 ≤ θ2 ≤ π. Find

x20 − x2

1 − x22 − x2

3.

Extend the transformation to Rn.

Problem 19. A fixed charge Q is located on the z-axis with coordinatesra = (0, 0, d/2), where d is interfocal distance of the prolate spheroidal

Functions 47

coordinates

x(η, ξ, φ) =12d((1− η2)(ξ2 − 1))1/2 cosφ

y(η, ξ, φ) =12d((1− η2)(ξ2 − 1))1/2 sin(φ)

z(η, ξ, φ) =12dηξ

where −1 ≤ η ≤ +1, 1 ≤ ξ ≤ ∞, 0 ≤ φ ≤ 2π. Express the Coulombpotential

V =Q

|r− ra|in prolate spheroidal coordinates.

Problem 20. Toroidal coordinates are defined by

x1(α, β, φ) =c sinh(α) cos(φ)

cosh(α)− cos(β)

x2(α, β, φ) =c sinh(α) sin(φ)

cosh(α)− cos(β)

x3(α, β, φ) =c sin(β)

cosh(α)− cos(β).

Use the L’Hosptial rule to find x1, x2, x3 for α→ 0 and β → 0.

Problem 21. Let c, θ ∈ R and

f(θ) = c(eiθ + e−iθ).

Calculate exp(f(θ)).

Problem 22. A continuous function f : R2 → R is called an alternatingfunction if

f(x, y) = −f(y, x).

Give an example of an analytic alternating function. Find the minima andmaxima of the function.

Problem 23. Consider the functions

j1(x) =1x2

(sin(x)− x cos(x))

j2(x) =1x3

((3− x2) sin(x)− 3x cos(x)).


Use the L’Hospital rule to find j1(0) and j2(0).

Problem 24. Find the invariance group of the function f : R→ R

f(x) = sin(x).

Problem 25. Let a > 0. Consider the transformation

u(x, y) =a sin(2ax)

2(sin2(ax) + sinh2(ay)), v(x, y) =

a sinh(2ay)2(sin2(ax) + sinh2(ay))

.

Find the inverse transformation.

Problem 26. (i) Find an analytic function f : R → R that has no fixedpoint and no critical point. Draw the function.(ii) Find an analytic function f : R→ R that has no fixed point and exactlyone critical point at x = 0. Draw the function.

Problem 27. Let f : R → R be an analytic function. Calculate thecommutator [

cos(x)d

dx, sin(x)

d

dx

]f.

Problem 28. (i) Consider the analytic function f : R2 → R2

f1(x1, x2) = sinh(x2), f2(x1, x2) = sinh(x1).

Show that this function admits the (only) fixed point (0, 0). Find thefunctional matrix at the fixed point(

∂f1/∂x1 ∂f1/∂x2

∂f2/∂x1 ∂f2/∂x2

)∣∣∣∣(0,0)

.

(ii) Consider the analytic function g : R2 → R2

g1(x1, x2) = sinh(x1), g2(x1, x2) = − sinh(x2).

Show that this function admits the (only) fixed point (0, 0). Find thefunctional matrix at the fixed point(

∂g1/∂x1 ∂g1/∂x2

∂g2/∂x1 ∂g2/∂x2

)∣∣∣∣(0,0)

.

(iii) Multiply the two matrices found in (i) and (ii).

Functions 49

(iv) Find the composite function h : R2 → R2

h(x) = (f g)(x) = f(g(x)).

Show that this function also admits the fixed point (0, 0). Find the func-tional matrix at this fixed point(

∂h1/∂x1 ∂h1/∂x2

∂h2/∂x1 ∂h2/∂x2

)∣∣∣∣(0,0)

.

Compare this matrix with the matrix found in (iii).

Problem 29. An approximation of e−x (x ≥ 0) as a rational polynomialusing a 3rd order Pade approximation is given by

e−x ≈ 1− x/2 + x2/10− x3/1201 + x/2 + x2/10 + x3/120

.

Note that e−x ≥ 0 for all x. For which value of x > 0 does the right-handside takes negative values?

Problem 30. Let x ∈ R. Consider the function

f(x) =x− ix+ i

.

Find f(0), f(1), f(−1) and f(x→∞). Is there an inverse?

Problem 31. Consider the function

f(x) = x3 − x2.

Find an approximation of the derivative of f by using

f ′(x) ≈ f(x− 2h)− 8f(x− h) + 8f(x+ h)− f(x+ 2h)12h

− 130h4f (5)(ξ)

f ′(x) ≈ −3f(x− h)− 10f(x) + 18f(x+ h)− 6f(x+ 2h) + f(x+ 3h)12h

+120h4f (5)(ξ)

f ′(x) ≈ −25f(x) + 48f(x+ h)− 36f(x+ 2h) + 16f(x+ 3h)− 3f(x+ 4h)12h

−15h4f (5)(ξ)

where x ≤ ξ ≤ x+ h.

Problem 32. Let z ∈ C. The Airy functions Ai(z) and Bi(z) are definedas sums of the power series

Ai(z) = c1f(z)− c2g(z)

Bi(z) =√

3(c1f(z) + c2g(z))


where

f(z) = 1 +13!z3 +

1 · 46!

z6 +1 · 4 · 7

9!z9 + · · ·

g(z) = z +24!z4 +

2 · 57!

z7 +2 · 5 · 8

10!z10 + · · ·

with

c1 =1

2πΓ(

13

)3−1/6, c2 =

12π

Γ(

23

)31/6.

Show that the radius of convergence of these infinite series is infinite.

Problem 33. Find the inverse of the function y = tanh(x). Note thaty ∈ (−1, 1).


f(x) =∞∑j=0

xj

(j + 1)!≡ ex − 1

x.

Find the fixed points and critical points of the function. Note that f(0) = 1.

Problem 35. Find non-negative analytic functions f : R→ R such that

f(0) = 0, f(1) = 1, f(2) = 0.

Problem 36. The Euler dilogarithm function Li2(x) is defined for x ∈(0, 1) as

Li2(x) =∞∑j=1

xj

j2= −

∫ x

0

ln(1− t)t

dt.

It can be analytically continued to the complex plane with the branch cutfrom 1 to ∞ along the real axis using the integral. Calculate Li2(1/2).

Problem 37. Let Ai be the Airy function and Jn, In are the Bessel andmodified Bessel functions of order n. Show that

d2Ai

dx2= xAi(x)

Ai(x) =13x1/2(I−1/3(z)− I1/3(z))

Ai(−x) =13x1/2(J1/3(z) + J−1/3(z))

Functions 51

where z := 2x3/2/3.

Problem 38. Find a continuous function f : R→ R such that

f(x) = f(−x), f(x) = f(x+ 2π), f(0) = 0.

Problem 39. Let N be an integer and N ≥ 2. The generalized hyperbolicfunction for a given N is defined as

f(N)j (x) :=

∞∑k=0

xj+kN

(j + kN)!, j = 0, 1, . . . , N − 1.

The functions f (N)j are analytic.

(i) Show thatf

(N)j (x) = f

(N)j+N (x)

i.e. f (N)j is periodic with respect to the index j.

(ii) Show thatd

dxf

(N)j (x) = f

(N)j−1(x).

(iii) Let ωN = 1, i.e. ω is the N -th primitive root of unity. Show that

f(N)j (x) =

1N

N−1∑k=0

ω−jk exp(ωkx).

Problem 40. Let n = 2, 3, 4, . . .. Find

Γ(n/2)Γ(n)

where Γ denotes the gamma function.

Problem 41. Consider the hyperspherical coordinates

x1(θ, φ, ψ) = cos(θ)x2(θ, φ, ψ) = sin(θ) cos(φ)x3(θ, φ, ψ) = sin(θ) sin(φ) cos(ψ)x4(θ, φ, ψ) = sin(θ) sin(φ) sin(ψ)

with x21+x2

2+x23+x2

4 = 1. Show that the angular distance can be calculatedas

cos(djk) =


cos(θj) cos(θk)+sin(θj) sin(θk)(cos(φj) cos(φk)+sin(φj) sin(φk) cos(ψj−ψk)).


f(x) = 1 + x+ 2x2 + 3x3.

Find the second order derivative of f at a = 1 applying

f ′′(a) = limε→0

f(a+ ε)− 2f(a) + f(a− ε)ε2

.

Problem 43. Let −1 < x < 1 and∞∑j=0

ajxj =

1√1− x

.

Find the expansion coefficients aj .

Problem 44. Show that the function

f(x) = cos(x) cosh(x) + 1

has infinitely many roots, i.e. solutions of f(x) = 0. What happens for xlarge?

Problem 45. The Jacobi elliptic functions sn(x, k), cn(x, k), dn(x, k)with k ∈ [0, 1] and k2 + k′2 = 1 have the properties

sn(x, 0) = sin(x), cn(x, 0) = cos(x), dn(x, 0) = 1

and

sn(x, 1) =ex − e−x

ex + e−x, cn(x, 1) =

2ex + e−x

= dn(x, 1).

We define

u1(x, y, k, k′) = sn(x, k)dn(y, k′)u2(x, y, k, k′) = cn(x, k)cn(y, k′)u3(x, y, k, k′) = dn(x, k)sn(y, k′).

(i) Find u1(x, y, 0, 1), u2(x, y, 0, 1), u3(x, y, 0, 1) and calculate u21(x, y, 0, 1)+

u22(x, y, 0, 1) + u2

3(x, y, 0, 1).(ii) Find u1(x, y, 1, 0), u2(x, y, 1, 0), u3(x, y, 1, 0) and calculate u2

1(x, y, 1, 0)+u2

2(x, y, 1, 0) + u23(x, y, 1, 0).

Functions 53

Problem 46. Let m be an integer with m ≥ 1. Then the gamma functionΓ is given by

Γ(m) = (m− 1)!.

Thus Γ(1) = Γ(2) = 1, Γ(3) = 2, Γ(4) = 6, Γ(5) = 24. Furthermore

Γ(m+12

) =1 · 3 · 5 · · · · · (2m− 1)

2m√π.

Thus Γ(3/2) =√π/2, Γ(5/2) = 3

√π/4, Γ(7/2) = 15

√π/8, Γ(9/2) =

105√π/16. Calculate

Γ(n2 −

12

)Γ(n2

)for m = 5, m = 10, m = 20.

Problem 47. Let c1 > 0, c2 > 0. Consider the function f : R→ R

f(x) = x2 +c1x

2

1 + c2x2.

Find the minima and maxima. Find the fixed points.

Problem 48. Let c > 0. Consider the function f : R→ R

fc(x) =4x2

sin2(cx

2

).

Find fc(0). Show that fc has a maximum at x = 0.

Problem 49. (i) Find the minima and maxima of the analytic functionf : R→ R

f(x) = cosh(x) cos(x).

Find the fixed points.(ii) Find the minima and maxima of the analytic function f : R→ R

f(x) = sinh(x) sin(x).

Find the fixed points.

Problem 50. Let S1 and S2 be two finite sets with n1 and n2 elements,respectively. The number of functions f : S1 → S2 is given by

nn12 .

(i) Let n1 = n2 = 2. Find all possible functions.


(ii) Let n1 = n2 = n. Then there are n! one-to-one functions f : S1 → S2.Let n = 3. Find all one-to-one functions.

Problem 51. Let µ > 0 and

R =

x1

x2

x3

, R′ =

x′1x′2x′3

.

(i) Show thatexp(−|R−R′|/µ)

|R−R′|=

∞∑n=0

εn cos(n(φ−φ′))∫ ∞

0

1k2 + 1/µ

Jn(kr)J(kr′) exp(−√k2 + 1/µ2|x3−x′3|)kdk

where εn = 1 for n = 0 and εn = 2 for n > 0 and Jm(kr) is ordinary Besselfunction of order m.(ii) Consider the functions

fs,k,n(R) =

√k

2πJn(kr)einφ+isx3

where 0 ≤ k <∞, −∞ < s <∞, and n = 0,±1,±2, . . .. Show that∫ 2π

0

∫ +∞

−∞dx3

∫ ∞0

fs,k,n(R)fs′,k′,n′(R)rdr = δnn′δ(s− s′)δ(k − k′).

Problem 52. Let f1, f2 : R→ R be analytic functions and

g1(x) =12

(f1(x) + f2(x)− |f1(x)− f2(x)|

g2(x) =12

(f1(x) + f2(x) + |f1(x)− f2(x)|.

Let f1(x) = sin(x) and f2(x) = cos(x). Find the maxima and minima ofg1. Find the maxima and minima of g2.

Problem 53. Let the function f be continuous on a closed interval [a, b]and is continuous differentiable in the open interval (a, b). Then there existsa point c in (a, b) such that (mean value theorem)

f(b)− f(a) = f ′(c)(b− a).

Apply the theorem to the function

f(x) = 1 + 2x+ 3x2

Functions 55

and the interval [−1, 1].

Problem 54. The plane (x1, x2, x3) : x1 + 12x2 + 1

3x3 = 1 and thecylinder (x1, x2, x3) : x2

1 + x22 = 4 intersect. Find the shortest distance

from (0, 0, 0) to this curve.

Problem 55. Consider the plane (x1, x2, x3) : 2x1 + 3x2 − x3 = 5 inthe three dimensional Euclidean space E3. Find the shortest distance from(0, 0, 0) to this plane.

Problem 56. We denotes by Q, R and C the rational, real and complexfields, respectively. Let B1 and B2 be Banach spaces. A map f : B1 → B2

is said to be additive if and only if

f(x1 + x2) = f(x1) + f(x2)

for all x1, x2 ∈ B1. Show that f is additive implies that f(qx) = qf(x) forall q ∈ Q. Show that f is additive and continuous implies that f(rx) =rf(x) for all r ∈ R.

Problem 57. Let a1d1 − b1c1 6= 0, a2d2 − b2c2 6= 0 and the functionsf1 : C→ C

f1(z) =a1z + b1c1z + d1

, f2(z) =a2z + b2c2z + d2

.

Find f2 f1, i.e. the function compositon.

Problem 58. Let x1, x2 ≥ 0. Consider

f(x1, x2) =√

1 + x1√1 + x1

√1 + x2

=∞∑j1=0

∞∑j2=0

F (j1, j2)xj11 xj22 .

Find the expansion coefficients.


f(x) =3x3

(sin(x)− x cos(x)).

Find f(0) using the L’Hospital rule.

Problem 60. Let

f(x) =sin(x)sinh(x)

.


Find f(0).

Problem 61. Consider the function f : R→ R

f(x) = x3.

Show that f is not convex. Show that f is convex if we restrict to thedomain x ≥ 0.

Problem 62. Consider the convex functions f : R → R and g : R → R.Show that f + g is convex. Show that max f, g is convex.

Problem 63. Definition (Convex Set). A subset C of Rn is said tobe convex if for any a and b in C and any θ in R, 0 ≤ θ ≤ 1, the n-tupleθa + (1− θ)b also belongs to C. In other words, if a and b are in C then

θa + (1− θ)b : 0 ≤ θ ≤ 1 ⊂ C.

Definition (Convex Polyhedron). Let a1, . . . , ap be p points in Rn. Then-tuple

p∑j=1

θjaj , θj ≥ 0, j = 1, . . . , p,p∑j=1

θj = 1

is called a convex combination (or a convex sum) of a1, . . . ,ap. If X ⊂ Rnthen the set of all (finite) convex combination of points of X is called theconvex hull of X and is denoted by H(X). If X is finite, X = a1, . . . ,ap,then H(X) is called the convex polyhedron spanned by a1, . . . ,ap and isalso denoted by H(a1, . . . ,ap).

Defintion. Let S be a nonempty convex set in Rn, where Rn is the n-dimensional Euclidean space. The function f : S → R is said to be convexif

f(λx1 + (1− λ)x2) ≤ λf(x1) + (1− λ)f(x2)

for each x1,x2 ∈ S and for each λ ∈ [0, 1]. The function is said to bestrictly convex if the above inequality holds as a strict inequality for eachdistinct x1,x2 ∈ S and for each λ ∈ (0, 1).

Leta1 = (1,−1), a2 = (2, 2), a3 = (3, 1)

be points of R2. Find the convex polyhedron.

Problem 64. Show that the function φ : [0,∞)→ R defined by

φ(x) =

0 if x = 0x lnx ifx 6= 0 (1)

Functions 57

is strictly convex, i.e.,

φ(αx+ βy) ≤ αφ(x) + βφ(y) (2)

if x, y ∈ [0,∞), αβ ≥ 0, α+ β = 1 with equality only when x = y or α = 0or β = 0. By induction we get

φ

(k∑i=1

αixi

)≤

k∑i=1

αiφ(xi) (3)

if

xi ∈ [0,∞), αi ≥ 0,k∑i=1

αi = 1. (4)

Equality holds only when all the xi, corresponding to non-zero αi, are equal.

Problem 65. Let x ≥ 0 and

f(x) = x ln(x)− x+ 1. (1)

(i) Show thatf(x) ≥ 0. (2)

(ii) Show that from L’Hospital rule we find that f(0) = 1.(iii) Show that the function is convex.

Problem 66. Show that f : R→ R

f(x) = |x| (1)

is convex.

Problem 67. Let f : R2 → R be defined as

f(x, y) = ax2 + by2 + 2cxy + d

where a, b, c, d ∈ R. For what values of a, b, c, d is f concave?

Problem 68. Let r1, r2 ∈ R. Consider the map f : R2 → R2

f1(x1, x2) = r1x1(1− x1 − x2), f2(x1, x2) = r2x1x2.

Find df1, df2, df1 ∧ df2.

Problem 69. Find x = arctan(−1/√

3).


Problem 70. Let f, g : R→ R be analytic functions. Find

ef(x) d2

dx2(e−f(x)g(x)).


f1(x1, x2) = sin(x21 + x2

2), f2(x1, x2) = cos(x21 + x2

2).

Find the Jacobian matrix and the Jacobian determinant.

Problem 72. Consider the map f : R→ R

f(x) = −12x2 − x+

12.

Then f(1) = −1, f(−1) = 1. Thus we have a periodic orbit with period 2.Is the orbit attracting or repelling? We set x1 = 1, x2 = −1. We have totest

|(df(x = x1)/dx)(df(x = x2)/dx)| < 1

for attracting and

|(df(x = x1)/dx)(df(x = x2)/dx)| > 1

for repelling.

Problem 73. Let α ∈ R and f(x) = eαx. Find

1h2

(f(x+ h)− 2f(x) + f(x− h)).

Then consider h→ 0.

Problem 74. Consider the analytic function f : R → R, f(x) = cos(x).The equation cos(x) = x has one solution, i.e. we have one fixed pointfor f . Consider f(f(x)) = cos(cos(x)). Does the equation cos(cos(x)) = xadmits other solutions besides the one of cos(x) = x.

Problem 75. Let a, b ∈ N. Consider the 2× 2 matrix

C =(

1 ab 1 + ab

)with det(C) = 1. Consider the map (generalized Arnold cat map)(

x1,τ+1

x2,τ+1

)= C

(x1,τ

x2,τ

)mod 1

Functions 59

where τ = 0, 1, 2, . . .. Find the eigenvalues of C which relates to the one-dimensional Liapunov exponents.

Problem 76. Consider the functions f : R→ R, g : R→ R

f(x) = 2x, g(x) = 3x2 − 1.

Find (f g)(x) = f(g(x)), (g f)(x) = g(f(x)) and the extrema of thesefunctions.

Problem 77. Let a ∈ R. Given the map f : R2 → R2

f(x1, x2) = (x1 + x2, 2x1 + ax2).

(i) Find the fixed points of the map.(ii) Find the matrix Df(x1, x2) and show that the matrix Df(x1, x2) isinvertible if and only if a 6= 2.

Problem 78. Let A, B be non-empty sets. If f : A → B is (1,1) andonto (i.e. bijective) one can define the inverse function f−1 : B → A as theunique function from B to A such that

(f f−1)(y) = y for all y ∈ B

(f−1 f)(x) = x for all x ∈ A.

The function f : (0, 1)→ R

x 7→ f(x) =x− 1/2x(x− 1)

is bijective. Find f−1. Note that f(1/2) = 0, f(1/4) = 4/3, f(3/4) = −4/3.

Problem 79. Give non-trivial functions f : R→ R which have no criticalpoints and satisfiy f(0) = 0. Trivial means f(x) = cx, where c is a constant.


Problem 1. Let f1, f2, f3 : R3 → R be continuously differentiable func-tion. Find the determinant of the 3× 3 matrix A = (ajk)

ajk :=∂fj∂xk− ∂fk∂xj

.


Apply computer algebra.

Problem 2. Consider the map f : R→ R given by

f(x) = x2 + x+ 1.

Find the minima and maxima of

g(x) = f(f(x))− f(x)f(x).

Apply computer algebra.

Problem 3. Consider the polynomials

f(x) = x2 + 2x+ 1, g(x) = x+ 4;

Find (f g)(x) = f(g(x)), (g f)(x) = g(f(x)) and f(g(x))− g(f(x)).

Problem 4. Let f : R → R be an analatic function. The Schwarzianderivative is defined by

S(f(x)) :=f ′′′(x)f ′(x)

− 32

(f ′′(x)f ′(x)

)2

where ′ denotes the derivative with respect to x. Let a, b, c, d ∈ R withad− bc 6= 0 and

g(x) =af(x) + b

cf(x) + d.

Show that S(g(x)) = S(f(x)). Apply computer algebra.


Problem 1. Find

limx→0

√1 + x−

√1− x

x.

Problem 2. Consider the analytic function f : R2 → R

f(x, y) =xex(ey − 1)− yey(ex − 1)

xy(ex − ey).

Functions 61

(i) Show that f(0, 0) = 12 applying the L’Hospital rule.

(ii) Show that

f(x, x) =ex − x− 1

x2.

(iii) Show that

f(x,−x) =tanh(x/2)

x.

Problem 3. Consider the map from R4 to the vector space of 2 × 2matrices over R

f(x1, x2, x3, x4) =(x1 + x4 x2 + x3

−x2 + x3 x1 − x4

).

Find the inverse of the map. Note that the determinant of the matrix isgiven by x2

1 + x22 − x2

3 − x24.

Problem 4. Let

D = (x1, x2, x3) : x21 + x2

2 + x23 = 1, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 .

Let v be a normalized vector in R3 with nonnegative entries and A be a 3×3matrix over R with strictly positive entries. Show that the map f : D → D

f(v) =Av‖Av‖

has a fixed point, i.e. there is a normalized vector v0 such that

Av0

‖Av0‖= v0.

Problem 5. Let f, g : R→ R be analytic functions. Find

(ef(x) d2

dx2e−f(x))x.

Problem 6. Let Z be the set of integers and N0 be the set of naturalnumbers including 0. Consider the one-to-one map f : Z→ N0

f(j) =

0 if j = 02j − 1 if j is positive−2j if j is negative

.


Give a C++ implementation with the class Verylong of the function f andits inverse.

Problem 7. Let f1, f2, . . . , fn be a set of convex functions from Rn →R. Show that the nonnegative linear combination

f(x) = α1f1(x) + α2f2(x) + · · ·+ αnfn(x), α1, α2, . . . , αn ≥ 0

is convex.

Problem 8. Study the map f : R2 → R2

f1(x1, x2) =√

1− sin(x2), f2(x1, x2) =√

1 + sin(x1).

First find the fixed points if there are any.

Problem 9. Let k ∈ N and x ≥ 0. Find the fixed points of the map

fk(x) =√k + x

and study their stability.

Problem 10. Consider the continuous function f : R+ → R+

f(x) =

2x x < 1max4− 2x, 1

4 x ≥ 1

Find the fixed points and study their stability.

Problem 11. Let

D = (x1, x2, x3) : x21 + x2

2 + x23 = 1, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

Let v be a normalized vector in R3 with nonnegative entries and A be a3 × 3 matrix over R with strictely positive entries. Show that the mapf : D → D

f(v) =Av‖Av‖

has fixed point, i.e. there is a v0 such that

Av0

‖Av0‖= v0.

Problem 12. Let f, g : R → R be two analytic functions and S be theSchwarzian derivative. Show that if Sf < 0 and Sg < 0, then

S(f g) < 0.

Functions 63

Problem 13. Let x, y ∈ R+. Consider the function f : (R+)2 → R

f(x, y) =√

1 + x√1 + x+

√1 + x

.

Then f(0, 0) = 1/2 and f(x, y) + f(y, x) = 1. Set

√1 + x√

1 + x+√

1 + x=∞∑m=0

∞∑n=0

F (m.n)xmyn.

Show that F (m,n) = −F (n,m) unless m = n = 0 and F (m,m) = 0 form ≥ 1. Show that

F (0, 0) = 1/2, F (0, 1) = −F (1, 0) = −1/8, F (0, 2) = −F (2, 0) = 1/16,

F (0, 3) = −F (3, 0) = −5/128, F (1, 2) = −F (2, 1) = −1/128.


f1(x1, x2) = ex1 cos(x2), f2(x1, x2) = ex1 sin(x2).

Find the fixed points of the map.

Problem 15. Consider the function f : R3 → R3

f(x) =

f1(x1, x2, x3)f2(x1, x2, x3)f3(x1, x2, x3)

=

x1 cos(x2) sin(x3)x1 sin(x2) sin(x3)

x1 cos(x3)

.



f(x) =(f1(x1, x2)f2(x1, x2)

)=(x2

1 − x22

2x1x2

).

The Jacobian matrix is(∂f1/∂x1 ∂f1/∂x2

∂f2/∂x1 ∂f2/∂x2

)=(

2x1 −2x2

2x2 2x1

)with determinant equal to 4(x2

1 +x22). Find the fixed points and study their

stability.


Problem 17. Consider the differential operators

D1 =d2

dx2+

c

x2+x2

16

D2 = sinh(α)(d2

dx2+

c

x2− x2

16

)+ cosh(α)

(− i

2xd

dx− i

4

)D3 = cosh(α)

(d2

dx2+

c

x2− x2

16

)+ sinh(α)

(− i

2xd

dx− i

4

).

(i) Show the differential operator satisfy (Lie algebra su(1, 1))

[D1, D2] = −iD3, [D2, D3] = iD1, [D3, D1] = iD2.

For the actual calculations let f : R → R be a smooth function and oneshows that [D1, D2]f(x) = −iD3f(x) etc.(ii) Show that

D23 −D2

1 −D22 = − 3

16− c

4.

Problem 18. Let

v(τ) =

v1(τ)v2(τ)v3(τ)

, w(τ) =

w1(τ)w2(τ)w3(τ)

where vj(τ), wj(τ) (j = 1, 2, 3) are smooth functions of τ . Calculate

d

dτ(v × (v ×w)).

Problem 19. Show that the (2, 2) Pade approximant of the cosine func-tion is given by

cos(x) ≈ 12− 5x2

12 + x2.

Problem 20. Let f(x) = x2 and g(x) = x ln(x). Show that

limx→∞

g(x)f(x)

= 0.

Problem 21. Let c > 0. Find

limx→0

sinh(cx)x

, limx→0

x

sinh(cx).

Functions 65

Problem 22. Show that the function f : (0, 1)→ R

f(x) =x− 1/2x(x− 1)

is bijective.

Problem 23. Show that the function

f(x) =π

sin(πx)

has poles at 0,±1,±2, . . ..

Problem 24. For any real number x one defines bxc (floor of x) as thelargest integer less than or equal to x. One defines dxe (ceiling of x) asthe smallest integer greater than or equal to x. Find the minima and themaxima of the function

f(x) = 2x− bxc.

Problem 25. Let 0 ≤ t < 2. Show that

ln(

2 + t

2− t

)= ln

(1 + t/21− t/2

)= 2arctanh(t/2).

Problem 26. Consider the continuous function f : R→ R

f(x) =12

(x

|x|+ 1+ 1).



f1(x1, x2) =12

(x21 − x2

2), f2(x1, x2) = x1x2.

Find the fixed points and study their stability. Is their an inverse f−1.

Problem 28. Explain

sin(α/2) = ±√

1− cos(α)2

, cos(α/2) = ±√

1 + cos(α)2

.


Problem 29. Consider the function f : R→ R defined by

f(x) =e−1/x2

for x 6= 00 for x = 0

Show that the function is not analytic. Note that f(x) = f(−x) and

df

dx=

2x3f(x),

d2f

dx2=(

4x6− 6x4

)f(x)

Furthermore limx→±∞ f(x) = 1.

Problem 30. Let N > 0. Show that

limε→0

ε−Ne−(1/ε) = 0.

Problem 31. Consider the map f : R2 → R2 given by(f1(x1, x2)f2(x1, x2)

)=(x1 cos(x2)x1 sin(x2)

).

Find the fixed points and study their stability. Show that the functionalmatrix is given by(

∂f1/∂x1 ∂f1/∂x2

∂f2/∂x1 ∂f2/∂x1

)=(

cos(x2) −x1 sin(x2)sin(x2) x1 sin(x2)

).

Is the map f invertible?


limx→0

sinh(2ax)sinh(x)

= 2a.

Problem 33. (i) Write down the first four terms of the Taylor expansionof cos(x). Then find (2, 2)[x] of the Pade approximant. Discuss the casex→∞ for both functions.(ii) Write down the first four terms of the Taylor expansion of cosh(x).Then find (2, 2)[2] of the Pade approximant. Discuss the case x → ∞ forboth functions.

Problem 34. Consider the Henon map (a > 0, b > 0)

x1,t+1 = a− bx2,t + x21,t, x2,t+1 = x1,t, t = 0, 1, 2, . . . .

Functions 67

Let |x1,0| < R, |x2,0| < R, where R is the larger root of λ2−(|b|+1)λ−a = 0.Show that all points (x1,0, x2,0) outside of this domain tend to ∞ or −∞for t→∞ or t→ −∞.

Problem 35. Consider the one-dimensional map (t = 0, 1, 2, . . .)

xt+1 =xt + 2xt + 1

, x0 ≥ 0.

Find the fixed points of the map. Let x0 = 0. Does limt→∞ xt tend to afixed point?

Problem 36. Consider the analytic functions f : R→ R, g : R→ R

f(x) = sinh(x), g(x) = 2x3.

Find f−1 g f , g−1 f g.

Problem 37. Let R > r > 0. Consider the function f : R2 → R2 (battleof the sexes)

f1(x1, x2) = (x1 1− x1 )(R 00 r

)(x2

1− x2

)= ((R+r)x2−r)x1+r(1−x2)

f2(x1, x2) = (x1 1− x1 )(r 00 R

)(x2

1− x2

)= ((R+r)x2−R)x2+R(1−x2).


Problem 38. Let c1, c2 ∈ R and f : R → R be an analytic function.Show that

exp(c1d

dx

)exp(c2x)f(x) ≡ exp(c2x) exp

(c1d

dx

)exp(c1c2)f(x).

Problem 39. Let n be a non-negative integer. Let k = 0, 1, . . . , n. Findthe derivative

∂n

∂xk∂xn−k(x+ y)n.

Show that∂n

∂xk∂xn−k(x+ y)n = n!.


Problem 40. The content (n-dimensional volume) bounded by a hyper-sphere of radius r is known to be

Vn =2rnπn/2

nΓ(n/2)

where Γ is the gamma function. Let r = 1. Show that

limn→∞

Vn = 0.

Problem 41. Let n ≥ 1. Show that the vector space spanned by

xn, yxn−1, . . . , yn−1x, yn

is n+ 1 dimensional.

Problem 42. Let ε, x ∈ R. Show that

limε→0

sinh(εx)sinh(ε)

= x, limε→0

sin(εx)sin(ε)

= x.

Problem 43. The Hurwitz zeta-function ζH(s, a) is defined by

ζH(s, a) :=∞∑k=0

(n+ a)−s, 0 < a ≤ 1, <(s) > 1.

(i) Show that the Hurwitz zeta-function can be written in the form

ζH(s, a) = a−s +1

Γ(s)

∞∑n=1

∫ ∞0

ts−1e−(n+a)tdt

where Γ(s) is the Gamma function.(ii) Show that

∂

∂aζH(s, a) =−sζH(s+ 1, a)

∂

∂sζH(s, a)

∣∣∣∣s=0

= ln(Γ(a))− 12

ln(2π).

Problem 44. Let f : R→ R be an analytic function. Show that (differ-ential identity)

dn

dxnf(x) = x−n

dn

dεnf(εx)

∣∣∣∣ε=1

.

Functions 69

Problem 45. Find an analytic function f : R→ R such that

f(∞) = 1, f(−∞) = 0

andf(x1) ≤ f(x2) whenever x1 ≤ x2.

Find the derivative of the function f and determine the maxima of thefunction df/dx.

Problem 46. Let

Θ(x) =

1 for x > 00 otherwise

Show that

Θ(x1 − x3)−Θ(x2 − x3) =

1 for x1 > x3 > x2

−1 for x2 > x3 > x1

0 otherwise

Problem 47. (i) Let

f(θ) =cos(π2 cos(θ)

)sin(θ)

.

Find f(θ) for θ = nπ (n ∈ Z) using the L’Hospital rule. Plot f(θ) as afunction of θ.(ii) Apply the L’Hospital rule to show that

limx→0

ex − 1x

= 1.

(iii) Show that

limx→0

sinh(x)x

= 1, limx→0

x

sinh(x)= 1.

(iv) Consider the function f : R→ R

f(x) =x cosh(x)− sinh(x)

2x sinh2(x).

Findlimx→0

f(x), limx→∞

f(x).


Problem 48. Consider the functions

f(x) =x

sinh(x), g(x) =

sinh(x)x

.

Findlimx→0

f(x), limx→0

g(x).

Problem 49. Let x ∈ R. Show that

limx→∞

1− e−x

x= 1.

Problem 50. (i) Show that

limx→0

1− cos(x)x

= 0.

(ii) Show that

limx→1

x− 1ln(x)

= 1.

(iii) Show that

limx→0

cosh(x)− cosh(2x)sinh(x)

= 0.

(iv) Let x > 0. Show that

limx→0

ln(1 + x)x

= 1.

Problem 51. Let α ∈ R. Find

limα→0

sin(α) cos(α)α

.

Problem 52. Let k > 0 and dimension 1/length. Find the extrema ofthe functions

f1(x) = ln(cosh(kx))f2(x) = cosh(kx) + α sinh(kx), −1 ≤ α ≤ +1

f3(x) =Asech(kx) ≡ A 1cosh(kx)

f4(x) = asech2(kx) ≡ a sinh2(kx)cosh2(kx)

.

Functions 71

Problem 53. Consider the analytic function f : R→ R,

f(x) = cos(x) + sin(x).

(i) Find the critical points of f and the minima and maxima of the function.(ii) Find the roots of f , i.e. solve f(x) = 0.(iii) Find the fixed points of f , i.e. solve f(x) = x.(iv) Find the differential equation together with the initial conditions thefunction f satisfies.

Problem 54. Consider the polynomial f : R→ R

f(x) = x3 − 32x2 + x− 3

2.

Show that there is a root, i.e. a solution of f(x) = 0 in the interval [1, 2].Apply the Newton method (say 3 steps) to find an approximate solutionfor the root. Apply the two initial condition x0 = 1 and x0 = 2. Afterfinding this root how would you proceed to find the other two roots?

Problem 55. A real valued function f defined on an open subset G ofa Banach space E is said to be Frechet differentiable at a point x ∈ G ifthere is f ′(x) ∈ E∗ such that

limv→0

|f(x+ v)− f(v)− 〈f ′(x), v〉|‖v‖

= 0.

Then f ′(x) is called the Frechet derivative of f at x. Let E = R. Letf(x) = x3 + 2x2 + 3x+ 4. Find the Frechet derivative.


f(x) =exp(−x2/2)

∫ x0

exp(τ2/2)dτx

.

Show thatlimx→0

f(x) = 1.

Problem 57. Consider the functions f : R→ [−1, 1], g : R→ [−1, 1]

f(x) = sin(x), g(x) = cos(x).

(i) Find the minima and maxima of the function

h1(x) = max(f(x), g(x)).


Is the function h1 differentiable?(ii) Find the minima and maxima of the function

h2(x) = min(f(x), g(x)).

Is the function h2 differentiable?

Problem 58. Let a ∈ R. Show that

limx→∞

(1 + a/x)x = ea.

Problem 59. Let k ∈ R, x, y ∈ +1,−1 such that xy = ±1. Show that

ekxy = cosh(k) + xy sinh(k).

Problem 60. Consider the coordinate transformation

x1(r, α, β, γ) = r sin(β/2) sin((γ − α)/2)x2(r, α, β, γ) = r sin(β/2) cos((γ − α)/2)x3(r, α, β, γ) = r cos(β/2) sin((γ + α)/2)x4(r, α, β, γ) = r cos(β/2) sin((γ + α)/2).

Show that x21 + x2

2 + x23 + x2

4 = r2.

Chapter 4

Polynomial

4.1 Solved ProblemsProblem 1. The Chebyshev polynomials are defined by

Tk(x) = cos(k arccos(x)), k = 0, 1, . . . x ∈ [−1, 1]

Thus the first six polynomials are

T0(x) = 1T1(x) = x

T2(x) = 2x2 − 1T3(x) = 4x3 − 3xT4(x) = 8x4 − 8x2 + 1T5(x) = 16x5 − 20x3 + 5x.

Find 1, x, x2, x3, x4, x5 as functions of T0, T1, T2, T3, T4, T5.

Problem 2. Let Ln(x), Hn(x) be the Laguerre polynomials and Hermitepolynomials, where n = 0, 1, . . .. Let

L(α)n (x) :=

x−αex

n!dn

dxn(e−xxn+α)

73


be the associated Laguerre polynomials with α > −1 and n = 0, 1, . . .. TheLaguerre polynomials are recovered by setting α = 0. We have

H2n(x) = (−4)nn!L(−1/2)n (x2) (1)

and the following addition formula for the associated Laguerre polynomialsLαn(x)

L(α+β+1)n (x+ y) =

n∑k=0

L(α)n−k(x)L(β)

k (y) . (2)

(i) Find a new sum rule by inserting (2) into (1).(ii) Consider the sum rule

1n!2n

Hn(√

2x)Hn(√

2y) =n∑k=1

(−1)kL(−1/2)n−k ((x+y)2)L(−1/2)

k ((x−y)2). (3)

Insert (1) into (3) to find a sum rule for Hermite polynomials.

Problem 3. The Hermite polynomial of degree n can be written as

Hn(x) =[n/2]∑k=0

(−1)kn!k!(n− 2k)!

(2x)n−2k.

Express xn using the Hermite polynomials.

Problem 4. Given the differentiable function f : R→ R with

f(x) = 4x(1− x).

(i) Find the fixed points of f(x) and f(f(x)).(ii) Find the critical points of f(x) and f(f(x)).

Problem 5. Let P (z) be a polynomial of degree n ≥ 2 with distinct zerosζ1, . . . , ζn. Show that

n∑j=1

1P ′(ζj)

= 0

where ′ denotes the derivative, i.e. P ′(ζ) ≡ dP (z = ζ)/dz.

Problem 6. Given a set of N real numbers x1, x2, . . . , xN . It is oftenuseful to express the sum of the j powers

sj = xj1 + xj2 + · · ·+ xjN , j = 0, 1, 2, . . .

Polynomial 75

in terms of the elementary symmetric functions

σ1 =N∑i=1

xi

σ2 =N∑i<j

xixj

σ3 =N∑

i<j<k

xixjxk

......

σN = x1x2 · · ·xN .

Consider the special case with three numbers x1, x2, x3. Then the elemen-tary symmetric functions are given by

σ1 = x1 + x2 + x3, σ2 = x1x2 + x1x3 + x2x3, σ3 = x1x2x3.

We know that the elementary symmetric functions are the coefficients (upto sign) of the polynomial with the roots x1, x2, x3. In other words thevalues of x1, x2, x3 each satisfy the polynomial equation

x3 − σ1x2 + σ2x− σ3 = 0. (1)

Find a recursion relation for

sj := xj1 + xj2 + xj3, j = 0, 1, 2, . . .

and give the initial values s0, s1, s2. Calculate s3 and s4.

Problem 7. The dominant tidal potential at position (r, φ, λ) due to themoon or sun is given by

U(r) =GM∗r2

r∗3P 0

2 (cos(ψ))

where M∗ is the mass of the moon or sun located at (r∗, φ∗, λ∗). Moreover,ψ is the angle between mass M∗ and the observation point at (r, φ, λ),where φ is the latitude and λ is the longitude. By the spherical cosinetheorem we have

cos(ψ) = sin(φ) sin(φ∗) + cos(φ) cos(φ∗) cos(λ− λ∗).

The Legendre polynomials are defined as

Pn(x) :=1

2nn!dn

dxn(x2 − 1)n


with n = 0, 1, 2, . . . and the associated Legendre polynomials are defined as

Pmn (x) := (1− x2)m/2dm

dxmPn(x) =

(1− x2)m/2

2nn!dm+n

dxm+n(x2 − 1)n

with P 0n(x) = Pn(x) and Pmn = 0 if m > n.

(i) Show that U(r) can be written as

U(r) =GM∗r2

r∗3

(P 0

2 (sinφ)P 02 (sinφ∗) +

13P 1

2 (sinφ)P 12 (sinφ∗) cos(λ− λ∗)

+112P 2

2 (sinφ)P 22 (sinφ∗) cos(2(λ− λ∗))

).

(ii) Give an interpretation (maxima and nodes) of the terms in the paren-thesis.

Problem 8. Consider the cubic equation

y3 + py + q = 0, p, q ∈ R, pq 6= 0. (1)

Show that applying the nonlinear transformation

y(z) := z − p

3z

equation (1) can be reduced to

z6 + qz3 − p3

27= 0

and with u = z3 to a quadratic equation.

Problem 9. Find all integers c for which the cubic equation

x3 − x+ c = 0

has three integer roots.

Problem 10. Let Φ be an endomorphism of the space Cn[X] of polyno-mials of degree n with complex coefficients, which maps a polynomial p(X)to the polynomial p(X+1). Let Ψ be an endomorphism of the space Cn[X]which maps a polynomial p(X) to (1−X)np

(X

1−X

), which of course is also

a polynomial. Show that we have a braid-like relation

Φ Ψ Φ = Ψ Φ Ψ.

Polynomial 77

Problem 11. Let a ∈ R. Let r and s be the roots of the quadraticequation

x2 + ax+a2 − 1

2= 0.

Find r3 +s3 in terms of a, and express it as a polynomial in a with rationalcoefficients.

Problem 12. Consider the polynomial p(x) = x3 − x2 + x − 2. Doesthere exist a nontrivial polynomial q(x) with real coefficients such that thedegree of every term of the product p(x)q(x) is a multiple of 3? If so, findone. If not, show there is none.

Problem 13. (i) Let n be a positive integer. Let f : [a, b] → R bea continuous function. The Bernstein polynomials of degree n associatedwith the continuous function f are given by

Bn(f(x), x) :=1

(b− a)n

n∑j=0

(n

j

)(x− a)j(b− x)n−jf(xj)

wherexj = a+ j

b− an

, j = 0, 1, . . . , n.

Consider the function f : [0, 1] → R given by f(x) = sin(4x). Show thatB2(f, x) is not a “good approximation” for f . Consider x = π/8.(ii) The Bernstein basis polynomials are defined as

Bn,j(x) :=(n

j

)xj(1− x)n−j , j = 0, 1, 2, . . . , n, x ∈ [0, 1].

Show thatn∑j=0

Bn,j(x) = 1.

Show that Bn,j satisfies the recursion relations

Bn,j(x) = (1− x)Bn−1,j(x) + xBn−1,j−1(x), j = 1, 2, . . . , n− 1Bn,0(x) = (1− x)Bn−1,0(x)Bn,n(x) = xBn−1,n−1(x).

Problem 14. A one-dimensional map f is called an invariant of a two-dimensional map g if

g(x, f(x)) = f(f(x)).


Letf(x) = 2x2 − 1.

Show that f is an invariant for

g(x, y) = y − 2x2 + 2y2 + d(1 + y − 2x2).

Problem 15. Consider the functions f(z) = z3 and h(z) = z+1/z. Finda function p such that

h(f(z)) = p(h(z)). (1)

Problem 16. Consider the map fc(z) = z2 + c, where c ∈ C. Find allcomplex c-values where the map fc has a fixed point z∗ with f ′c(z

∗) = −1.

Problem 17. Let p(x, y) be a real polynomial. Show that if p(x, y) = 0for infinitely many (x, y) on the unit circle x2 + y2 = 1, then p(x, y) = 0 onthe unit circle.

Problem 18. Show that the equation

zn = a (1)

where n is a positive integer and a is any nonzero complex number, hasexactly n roots. Hint. Set

a = ρ(cos(φ) + i sin(φ)). (2)

Problem 19. Let A be a 2 × 2 matrix over the real numbers R. Thetrace is defined as

tr(A) = a11 + a22.

It can be proved that the trace is the sum of the eigenvalues of A, i.e.

tr(A) = λ1 + λ2.

Thus we havetr(A2) = λ2

1 + λ22.

Let

A =(

1 11 1

).

Find the eigenvalues using the two equations.

Polynomial 79

Problem 20. Find the zeros of the cubic polynomial

α(x) := a0 + a1x+ a2x2 + x3 (1)

over C, where a0, a1, a2 ∈ R and a0 6= 0.

Problem 21. Show that the zeros of

z3 − 6z2 + 11z − 6 = 0 (1)

are given by 1, 2, and 3.

Problem 22. Find the zeros of the quartic polynomial

α(x) = a0 + a1x+ a2x2 + a3x

3 + x4 (1)

over C when a0 6= 0.

Problem 23. Find the zeros of

α(x) = 35− 16x− 4x3 + x4. (1)

Problem 24. The variational equation of the Lorenz model

dX

dt= −σX + σY (1a)

dY

dt= −XZ + τX − Y (1b)

dX

dt= XY − bZ (1c)

is given by dx0/dtdy0/dtdz0/dt

=

−σ σ 0(τ − Z) −1 −XY X −b

x0

y0

z0

. (2)

(i) Show that Lorenz model possess the steady-state solution X = Y =Z = 0, representing the state of no convection.(ii) Show that with this basic solution, the characteristic equation of thevariational matrix is

(λ+ b)(λ2 + (σ + 1)λ+ σ(1− τ)) = 0. (3)


(iii) Show that this equation has three real roots when τ > 0; all are negativewhen τ < 1, but one is positive when τ > 1. The criterion for the onset ofconvection is therefore τ = 1.(iv) Show that when τ > 1, system (1) possess two additional steady statesolutions

X = Y = ±√b(τ − 1), Z = τ − 1. (4)

(v) Show that for either of these solutions, the characteristic equation ofthe matrix in (2) is

λ3 + (σ + b+ 1)λ2 + (τ + σ)bλ+ 2σb(τ − 1) = 0. (5)

(vi) Show that this equation possesses one real negative root and two com-plex conjugate roots when τ > 1. Show that the complex conjugate rootsare pure imaginary if the product of the coefficients of λ2 and λ equals theconstant term, or

τ = σ(σ + b+ 3)(σ − b− 1)−1. (6)

Problem 25. The variational equation of the Lotka Volterra model

du1

dt= u1 − u1u2,

du2

dt= −u2 + u1u2 (1)

is given by (dv1/dtdv2/dt

)=(

1− u2 −u1

u2 −1 + u1

)(v1

v2

)(2)

where u1 > 0 and u2 > 0.(i) Show that (1) possess the steady-state solution u1 = u2 = 1.(ii) Show that with this basic solution, the characteristic equation of thematrix in (2) is

λ2 + 1 = 0. (3)

(iii) Find the solution of the characteristic equation and discuss.

Problem 26. Show the following. Let P (z) be a polynomial. Then either

1. P (z) has a fixed point q with P ′(q) = 1,2. P (z) has a fixed point q with |P ′(q)| > 1.

Problem 27. Let Sn be the symmetric group. The symmetric groupSn acts naturally on polynomials in n variables. For a polynomial p in nvariables, define the symmetrized polynomial associated to p, Symn(p), by

Symn(p) :=∑σ∈Sn

σ(p).

Polynomial 81

Let n = 2 andp(x1, x2) = x2

1x2 + 2x2.

Find Sym2(p).

Problem 28. Consider the system of equations

z1 + z2 + · · ·+ zn−1 + zn = 0z1z2 + z2z3 + · · ·+ zn−1zn + znz1 = 0

...z1z2 · · · zn−1 + z2z3 · · · zn + · · ·+ zn−1zn · · · zn−3 + znz1 · · · zn−2 = 0

z1z2 · · · zn = 1.

Find the solutions for the case n = 2 and n = 3. This system of equationsarise as follows. Let p be a prime number. A vector x = (x0, x1, . . . , xp−1) ∈Cp viewed as a function Z/p → C has discrete Fourier transform x =(x0, x1, . . . , xp−1), where

xj =p−1∑k=0

ωjkxk, ω := e2πi/p.

The vector x is called equimodular if all its coordinates have the same abso-lute value, and x is called bi-equimodular if both x and x are equimodular.The question is: Which vectors are bi-equimodular?

Problem 29. Let α ∈ R. Find the roots of the characteristic equation

λ6 − 2 cos(3α)λ3 + 1 = 0.

Problem 30. Show that the n-th order polynomial

p(x) = a0 + a1x+ a2x2 + · · ·+ anx

n

which goes exactly through n+ 1 data points is unique.

Problem 31. Let p : R→ R be a polynomial that satisfies

p(1− x) + 2p(x) = 3x

for all x ∈ R. Find the values of p(0) and p(1). Give an example of apolynomial that satisfies this condition.


Problem 32. (i) Let x1, x2 ∈ R and x1 6= x2. Find the solutions of thesystem of equations

x1 −1

(x1 − x2)2= 0, x2 +

1(x1 − x2)2

= 0.

(ii) Let x1, x2, x3 ∈ R and x1 6= x2, x1 6= x3, x2 6= x3. Find the solutionsof the system of equations

x1 −1

(x1 − x2)2− 1

(x1 − x3)2= 0

x2 +1

(x1 − x2)2− 1

(x2 − x3)2= 0

x3 +1

(x1 − x3)2+

1(x2 − x3)2

= 0.

Problem 33. Let z, w ∈ C. Find the solution of the system of equations

|z|2 + |w|2 = 1, z2 + w3 = 0.

Problem 34. Consider the two polynomials

p1(x) = a0 + a1x+ · · ·+ anxn, p2(x) = b0 + b1x+ · · ·+ bmx

m

where n = deg(p1) and m = deg(p2). Assume that n > m. Let r(x) =p2(x)/p1(x). We expand r(x) in powers of 1/x, i.e.

r(x) =c1x

+c2x2

+ · · ·

From the coefficients c1, c2, . . . , c2n−1 we can form an n×n Hankel matrix

Hn =

c1 c2 · · · cnc2 c3 · · · cn+1

......

. . ....

cn cn+1 · · · c2n−1

.

The determinant of this matrix is proportional to the resultant of the twopolynomials. If the resultant vanishes, then the two polynomials have anon-trivial greates common divisor. Apply this theorem to the polynomials

p1(x) = x3 + 6x2 + 11x+ 6, p2(x) = x2 + 4x+ 3.

Polynomial 83

Problem 35. Let p1 and p2 two polynomials with n = degree(p1), m =degree(p2) and n > m. We expand the rational function

r(x) =p2(x)p1(x)

with respect to powers of 1/x, i.e.,

r(x) = d1x−1 + d2x

−2 + · · ·

The coefficients d1, d2, . . . , d2n−1 are inserted into the n× n Hankel matrix

Hn =

d1 d2 . . . dnd2 d3 . . . dn+1

......

. . ....

dn dn+1 . . . d2n−1

.

A Hankel matrix is a diagonal matrix in which all the elements are thesame along any diagonal that slopes from northeast to southwest. If thedeterminant of this matrix is zero, then the two polynomials have a non-trival common divisor. Apply this algorithm to the polynomials

p1(x) = x3 + 6x2 + 11x+ 6, p2(x) = x2 + 6x+ 8

to test whether they have a common non-trivial divisor.

Problem 36. Let p be a polynomial with coefficients in R. If the equationp(x) = 0 has repeated roots, then p(x) and dp(x)/dx have a highest commonfactor. Apply this to the polynomial

p(x) = 32x4 − 64x3 + 24x2 + 8x− 3.

Problem 37. Let p be a polynomial with real coefficients. The equationp(x) = 0 cannot have more positive roots than there are changes of signfrom + to − and from − to + in the coefficients of the polynomial p(x),or more negative roots than there are changes of sign in p(−x) (Descartesrule of signs). Apply the rule to the polynomiol

p(x) = x6 + 7x3 + x− 2 = 0.

Problem 38. The Liouville-Riemann definition for the fractional integraloperator D−qx is given by

D−qx f(x) :=1

Γ(q)

∫ x

0

(x− y)q−1f(y)dy, q > 0.


The fractional differential operator Dνx for ν > 0 is given by the definition

Dνxf(x) :=

dn

dxn(Dν−n

x f(x)), ν − n < 0.

Let f(x) = x2. Find D−qx f(x).

Problem 39. Let n ∈ N. Show by induction that xn − yn is divisiblewithout remainder by x− y for all values of n. We have

xn+1 − yn+1 ≡ x(xn − yn) + yn(x− y).

Problem 40. Consider the polynomial

Pn(x) =n∑j=0

(−1)jajxn−j , a0 = 1.

The homogeneous product sums symmetric functions hk(x1, . . . , xn) of thezeros of this polynomial are defined as follows

n∏j=1

(1− xjx) = (1 + h1x+ h2x2 + h3x

3 + · · ·)−1.

Show that the first few sums are given explicitly by

h1(x1, . . . , xn) =n∑j=1

xj

h2(x1, . . . , xn) =n∑j=1

x2j +

n∑j<k

xjxk

h3(x1, . . . , xn) =n∑j=1

x3j +

n∑j 6=k

+n∑

j<k<`

xjxkx`.

Problem 41. Consider the quintic equation

p(x) = x5 − 5x3 + 5x− 5 = 0.

Is p irreducible over the rational numbers? What is the Galois group of p?Look for solutions of the form r + 1/r.

Problem 42. The Bernoulli polynomials Bn(x) (n = 0, 1, . . .) can bedefined recursively by

dBn(x)dx

= nBn−1, n = 1, 2, . . .

Polynomial 85

with B0(x) = 1 and the condition∫ 1

0

Bn(x)dx = 0, n ≥ 1.

The Bernoulli numbers Bn are defined by Bn := Bn(x = 0).(i) Find the first four Bernoulli polynomials.(ii) Show that

Bm(x) =m∑n=0

1n+ 1

n∑k=0

(−1)k(n

k

)(x+ k)m.

(iii) Show thatBn(1− x) = (−1)nBn(x).

Problem 43. Consider the operators

D1 = x, D2 =d

dx, D3 = x

d

dx

which apply to the function of the Bargmann space

B := f(n) =xn√n!, n ∈ N0, x ∈ R .

Show that

D1f(n) =√n+ 1f(n+ 1), D2f(n) =

√nf(n− 1), D3f(n) = nf(n).

Find the commutators [D1, D2], [D1, D3], [D2, D3].

Problem 44. Express the polynomial

p(x) = 7x21 + 6x2

2 + 5x23 − 4x1x2 − 4x2x3 + 14x1 − 8x2 + 10x3 + 6

in matrix form

p(x) = (x1 x2 x3 )A

x1

x2

x3

+ vT

x1

x2

x3

+ c

Find the eigenvalues and normalized eigenvectors of the 3× 3 matrix.

Problem 45. Find the polynomials generated by

dpj+1(x)dx

= (j + 1)pj(x), j = 0, 1, 2, . . .


with p0(x) = 1. The constants of integration we set to 0.

Problem 46. The complete Bell polynomials Bj(x1, x2, . . . , xj) are givenby the exponential generating function

exp

∞∑j=1

xjtj

j!

=∞∑n=0

Bn(x1, . . . , xn)tn

n!. (1)

Taking the n-th derivative with respect to t we obtain

dn

dtnexp

∞∑j=1

xjtj

j!

∣∣∣∣∣∣t=0

= Bn(x1, . . . , xn) (2)

with B0 = 1. Find the first four Bell polynomials.

Programming Problem

Problem 1. Consider the polynomials f1 : R→ R, f2 : R→ R

f1(x) = x2 + 2x+ 1, f2(x) = 2x.

Apply Maxima to find

h1(x) = f1(f2(x)), h2(x) = f2(f1(x))

and C = f1 − f2.


Problem 1. Construct a polynomial

p(x) = x2 + ax+ b

that admits the roots16

(√

3 + 3), −16

(√

3− 3)

and the fixed points

16

(√

30 + 6), −16

(√

30− 6).

Polynomial 87

Problem 2. Let a, b ∈ R. Consider the quartic equation

(λ− 1)2(λ− b2)2 − λ2a2(1− b2)2 = 0

Show that the roots are given by

λ1(a, b) =12

(1− a+ (1 + a)b2 + (T+)1/2)

λ2(a, b) =12

(1− a+ (1 + a)b2 − (T+)1/2)

λ3(a, b) =12

(1 + a+ (1− a)b2 + (T−)1/2)

λ4(a, b) =12

(1 + a+ (1− a)b2 − (T−)1/2)

where T± = (1 + a2)(1− b2)2 ± 2a(b4 − 1).

Problem 3. Let n be a positive integer. Show that xn − y2 has x− y asa factor for all n.

Problem 4. Let s = 1/2, 1, 3/2, 2, . . . be the spin values. The Brillouinfunction is defined as

Bs(x) :=2s+ 1

2scoth

(2s+ 1

2sx

)− 1

2scoth

(12sx

)Find Bs(x = 0) applying L’Hospital.

Problem 5. Let n ∈ N0 and x,m ∈ R. The Sonine polynomials aredefined by

Snm(x) =n∑j=0

(−1)jΓ(m+ n+ 1)xj

Γ(m+ j + 1)(n− j)!j!

where Γ denotes the gamma function. Show that

S0m(x) = 1, S1

m(x) = m+ 1− x.

Show that the Sonine polynomials satisfy∫ ∞0

e−xxmSjm(x)Skm(x)dx =Γ(m+ j + 1)

j!= δj,k.

Problem 6. (i) Show that the cubic roots of unity z3 = 1 are

1, w = −12

+√

3i2, w2 = −1

2−√

3i2. (2)


(ii) Show that the zeros of z3 + 1 = 0 are given by

−1,12

+i

2

√3,

12− i

2

√3. (1)

(iii) Show that the set

S = ω1 = −12

+i

2

√3, ω2 = −1

2− i

2

√3, ω3 = 1

of the cubic roots of 1, forms an abelian group with respect to multiplicationon the set of complex numbers C, i.e. show that

ω1ω2 = ω3, ω2ω1 = ω3, ω1ω3 = ω1,

ω3ω1 = ω1, ω2ω3 = ω2, ω3ω2 = ω2. (1)

Obviously, the neutral element is ω3. From (1) we see that each elementhas an inverse (which is unique), i.e.

(ω1)−1 = ω2, (ω2)−1 = ω1, (ω3)−1 = ω3.

The associative law is true for all complex numbers.

Problem 7. Let a, b ∈ R and z be denote a root of the quadratic equation

z2 + az + b = 0. (1)

Show that the sequence of powers of z, zn, n ≥ 2, satisfies the lineardifference equation

wn + awn−1 + bwn−2 = 0, n ≥ 2. (2)

Set wn = xn. Then wn−1 = xn−1.

Problem 8. A real number is said to be algebraic if it is a zero of apolynomial

cnxn + cn−1x

n−1 + · · ·+ c0

where the coefficients cj are integers. The height of such polynomials isdefined as the positive integer number

h = n+ |cn|+ |cn−1|+ · · ·+ |c0|.

Show that there are only finitely many polynomials of height h. Show thatthe set of all algebraic real numbers is enumerable.

Polynomial 89

Problem 9. Let p0 = 1, p1, p2, . . . be polynomials which satisfy therecursion relation

pn =1

2n

n∑k=1

22k

(n!

(n− k)!

)2

xkpn−k, n ∈ N.

Giva a SymbolicC++ and Maxima implementation of this recursion rela-tion. For example p1 = 2x1.

Problem 10. Show that every polynomial α(x) ∈ C[x] of degree m ≥ 1has precisely m zeros over C, where any zero of multiplicity is to be countedas n of the m zeros.

Problem 11. Show that if r ∈ C is a zero of any polynomial α(x) withreal coefficients, then r is also a zero of α(x), where r denotes the complexconjugate of r.

Problem 12. Let

P = fi(x), i = 1, . . . , k, x = (x1 . . . xn) (1)

be a set of multivariate polynomials in the ring Q[x1 . . . xn], with a solutionset

S = S(fi . . . fk) = x | fi(x) = 0, ∀i = 1, . . . , k (2)

All polynomialsu(x) =

∑j

gj(x)fj(x) (3)

for arbitrary polynomials gj will vanish in all points of S. The set of all uestablishes a polynomial ideal

I = I(f1 · · · fk) =

∑j

gj(x)fj(x)

(4)

and classical algebra tells that S is invariant if we replace the set f1, · · · , fkby any other basis for the ideal I(f1, · · · fk).

The Buchberger algorithm allows one to transform the set of polynomialsinto a canonical basis of the same ideal, the Grobner basis GB = GB(I).For the purpose of the equation solving especially Grobner bases computedunder lexicographical term ordering are important. They allow one to de-termine the set S directly. If I has dimension zero (S is a finite set ofisolated points, GB has in most cases the form

g1(x1, xk) = x1 + c1,m−1xm−1k + c1,m−2x

m−2k + · · ·+ c1,0


g2(x2, xk) = x2 + c2,m−1xm−1k + c2,m−2x

m−2k + · · ·+ c2,0

· · ·gk−1(xk−1, xk) = xk−1 + ck−1,m−1x

m−1k + ck−1,m−2x

m−2k + · · ·+ ck−1,0

gk(xk) = xmk + ck,m−1xm−1k + ck,m−2x

m−2k + · · ·+ ck,0.

A basis in this form has the elimination property: the variable dependencyhas been reduced to a triangular form, just as with a Gaussian eliminationin the linear case. The last polynomial is univariate in xk. It can besolved with usual algebraic or numeric techniques; its zero xk then arepropagated into the remaining polynomials, which then immediately allowone to determine the corresponding coordinates (x1, . . . , xk−1).Consider the system

y2 − 6y, xy, 2x2 − 3y − 6x+ 18, 6z − y + 2x. (4)

(i) Show that that this system has for x, y, z the lexicographical Grobnerbasis

g1(x, z) = x−z2+2z−1, g2(y, z) = y−2z2−2z−2, g3(z) = z3−1. (5)

(iii) Show that the roots of the third polynomial are given byz = 1, z =

1−√

3i2

, z =√

3i− 12

. (6)

If we propagate one of them into the basis, we generate univariate polyno-mials of degree one which can be solved immediately; e.g. Selecting 1−

√3i

2for z the basis reduces to

x =3√

3i+ 32

, y, 0

(7)

such that the final solution for this branch isx =

3√

3i+ 32

, y = 0, z =1−√

3i2

. (8)

For zero dimensional problems the last polynomial will always be univariate.However, in degenerate cases the other polynomials can contain their lead-ing variable in a higher degree, then containing more mixed terms with thefollowing variables. And there can be additional polynomials with mixedleading terms (of lower degree) imposing some restrictions. But the varibledependency pattern will remain triangular (evenutally with more than krows). There is a special algorithm for decomposing such ideals using idealquotients.

Chapter 5

Equations

5.1 Solved ProblemsProblem 1. Solve the equation

13

=1x

+14.

Problem 2. Let x1, x2 be positive real numbers. Consider the equation

x2 sin(θ) = x1 cos(θ).

Find sin(θ).

Problem 3. Let L be a given positive real number. Solve the system oftwo coupled nonlinear equations

1 = x2L+ 2x2y + Lx2y2

0 =−x2 + 2x2y + Lx2y2.

Problem 4. Let c, y ∈ R. Solve the quadratic equation

x = cy + yx− x2

91


with respect to x.

Problem 5. Consider a triangle in the plane. Consider a point withinthe triangle. We draw lines from this point to the three vertices, therebydividing the triangle into three triangles of area A1, A2, A3. The sides ofthe triangle are designated by the same number as the opposite vertex. Theareas are identified by the number of the adjacent side. The quantities Lj(j = 1, 2, 3)

L1 =A1

A, L2 =

A2

A, L3 =

A3

Awhere A is the area of the original triangle, are defined to be the triangularcoordinates. Show that

L1 + L2 + L3 = 1.

The relationship between the Cartesian coordinates x, y which are thecoordinates of the points in the elements, and the triangular coordinatesL1, L2, L3 are

x = L1x1 + L2x2 + L3x3, y = L1y1 + L2y2 + L3y3

where xj , yj (j = 1, 2, 3) are the coordinates of the nodes. The triangularcoordinates can be expressed in terms of the known locations of the vertices,i.e. 1

xy

=

1 1 1x1 x2 x3

y1 y2 y3

L1

L2

L3

.

Find L1, L2, L3 as function of xj , yj , x, y.

Problem 6. Let x, y 6= 0. Find all solutions of the equation

4x2y2

(x2 + y2)2= 1.

Problem 7. Solve the equation

71x2 = 133 (mod 11).

Problem 8. Let x, y ∈ N. Find all solutions of

16x+ 7y = 601.

Problem 9. Let x, y ∈ Z. Find all solutions of

18x+ 12y = 4.

Equations 93

Problem 10. Consider the two hyperplane (n ≥ 1)

x1 + x2 + · · ·+ xn = 2, x1 + x2 + · · ·+ xn = −2.

The hyperplanes do no intersect. Find the shortest distance between thehyperplanes. First consider the cases n = 1 and n = 2 and then the generalcase. What happens if n→∞?

Problem 11. Let d be a positive distance, v, V be velocities v 6= V andT1, T2 time-intervals. Assume that

d

V + v= T1,

d

V − v= T2.

Find d/V .

Problem 12. Solve the quadratic equation

x2 − ix− (1 + i) = 0.

Problem 13. Let z ∈ C. Solve(z + 1z − 1

)= i.

Problem 14. Solve the system of nonlinear equations

x2 − (y − z)2 = a2

y2 − (z − x)2 = b2

z2 − (x− y)2 = c2.

Problem 15. Show that the quartic equation

x4 + qx2 + rx+ s = 0

can be written as

(x2 − ex+ f)(x2 + ex+ g) = 0

where e2 is the root of the cubic equation

z3 + 2qz2 + (q2 − 4s)z − r2 = 0.


Problem 16. Let

z =(z1

z2

), w =

(w1

w2

)be elements of C2. Solve the equation z∗w = w∗z.

Problem 17. A special set of coordinates on Sn called spheroconical (orelliptic spherical) coordinates are defined as follows: For a given set of realnumbers α1 < α2 < · · · < αn+1 and nonzero x1, . . ., xn+1 the coordinatesλj (j = 1, . . . , n) are the solutions of the equation

n+1∑j=1

x2j

λ− αj.

Find the solutions for n = 2.

Problem 18. Find all solutions of the system of equations

12

1111

=

cos(α) cos(β)cos(α) sin(β)sin(α) cos(β)sin(α) sin(β)

.

Problem 19. Let φ ∈ [0, 2π). Solve the cubic equation

4x3 − 3x− cos(φ) = 0

over the real numbers.

Problem 20. Let ε > 0. Find the solution of the coupled two non-linearequations

εx2 + x− y − 1 = 0, εy2 + x− y − 1 = 0.

Study ε→ 0 for these solutions.

Problem 21. Consider the six 3× 3 matrices

X12 =

a12 b12 0c12 d12 00 0 1

, X ′12 =

a′12 b′12 0c′12 d′12 00 0 1

,

X13 =

a13 0 b13

0 1 0c13 0 d13

, X ′13 =

a′13 0 b′13

0 1 0c′13 0 d′13

,

Equations 95

X23 =

1 0 00 a23 b23

0 c23 d23

, X ′23 =

1 0 00 a′23 b′23

0 c′23 d′23

.

Find the 9 conditions on the entries such that (local Yang-Baxter equation)

X12X13X23 = X ′23X′13X

′12.

Find solutions of these 9 equations for the 24 unkowns a12, . . . , d′23.

Problem 22. Solve the equations

2 arcsin(x) + arcsin(2x)− π/2 = 0

2 arcsin(x)− arccos(3x) = 0

arccos(x)− arctan(x) = 0

arccos(2x2 − 4x− 2)− 2 arcsin(x) = 0

2 arctan(x)− arctan(

2x1− x2

)= 0.

Problem 23. Let x ∈ R. Find all values of x that satisfy

|5x+ 7| = 3.

Then find the smallest and largest values.

Problem 24. Let θ ∈ [0, 2π). Can one find x ∈ R such that

sin(θ) =e−x/2√1 + e−x

?

Problem 25. Let x < 1. Find a solution of the equation

f2(x)− 2f(x) + x = 0.

Problem 26. What are the conditions on c1 > 0 and c2 > 0 such thatthe system of equations

x1 + x2 = c1, x1x2 = c2

has real solutions?


Problem 27. Let m, n be positive integers. Find all solution of thesystem of equations

2mn+ (m2 + n2) = (m2 − n2)2, m− n = 1.

Problem 28. Let x ∈ R.(i) Find all solutions of 2x = |x|+ 1.(ii) Find all solutions of 2x = −|x|+ 1.

Problem 29. Let x ∈ Z. Solve the equation

x2 − 2x+ 2 = 0 mod 5.

Problem 30. Let a > b > 0. Find the points of intersections of the twoellipses

x21

a2+x2

2

b2= 1,

x21

b2+x2

2

a2= 1.

Problem 31. Let r1 ≥ 0, r2 ≥ 0, r3 ≥ 0. Find the solutions of thesystem of equations

r1 + r2 + r3 = 1, r21 + r2

2 + r23 = 1.

Problem 32. Consider the cubic equation

p(x) = x3 − 6x2 + 11x− 6 = 0.

Let x1, x2, x3 be the roots. Given the equations

x1 + x2 + x3 = −a1, x1x2 + x2x3 + x3x1 = a2, x1x2x3 = −a3

Find a1, a2, a3.

Problem 33. Let z ∈ C. Find all solutions of

z + z∗ + zz∗ = 0.

Set z = reiφ with r ≥ 0 and φ ∈ [0, 2π). An alternative would be settingz = x+ iy with x, y ∈ R.

Problem 34. We want to find the positive root of f(x) = 0 with

f(x) = x3 − x2 − x− 1.

Equations 97

We write x3 − x2 − x− 1 = 0 as x = 1 + 1/x+ 1/x2 and set

xt+1 = 1 + 1/xt + 1/x2t , t = 0, 1, . . .

with x0 = 1. Do we find the positive root?

Problem 35. Find all 2× 2 invertible matrices S over R with det(S) = 1such that

S

(0 10 1

)=(

0 10 1

)S S

(0 10 1

)S−1 =

(0 10 1

).

Thus we have to solve the three equations

s21 = 0, s11 + s12 = s22, s11s22 = 1.

Problem 36. Let z ∈ C. Find all solutions of

z + z∗ + zz∗ = 0.

Set z = reiφ with r ≥ 0.

Problem 37. Let a be a real constant. Solve the two equations

−a3 + x3 − 3ax2 + 3a2x1 = 0, x2 − 2ax1 + a2 = 0

with respect to x1, x2, x3.


Problem 1. Let ε ∈ R. Consider the quadratic equation

x2 − εx+ ε− 1 = 0.

Find the roots x1(ε), x2(ε). Then find the mimimum of x21 +x2

2 with respectto ε.

A SymbolicC++ program to solve this problem is:

// quadratic.cpp

#include <iostream>

#include "symbolicc++.h"

using namespace std;


int main(void)

Symbolic x("x"), eps("eps"), f = 0;

Equations soln = solve((x^2)-eps*x+eps-1==0,x);

cout << "Solutions: " << endl << soln << endl;

Equations::iterator i;

for(i=soln.begin();i!=soln.end();i++) f += (i->rhs^2);

cout << "f(eps) = " << f << endl;

Equations min = solve(df(f,eps)==0,eps);

for(i=min.begin();i!=min.end();i++)

if(double(df(f,eps,2)[*i]) > 0)

cout << "Minimum at " << *i << endl;

return 0;

/*

Solutions:

[ x == eps-1,

x == 1 ]

f(eps) = eps^(2)-2*eps+2

Minimum at eps == 1

*/


Problem 1. Let x, y ∈ R. Find all solutions of

e−x−y = e−x + e−y.

Problem 2. Solve the system of nonlinear equations

x1x2 = 1, x2x3 = 2, x3x1 = 3.

Extend to the general case

x1x2 = c1, x2x3 = c2, . . . , xn−1xn = cn−1, xnx1 = cn

where cj (j = 1, . . . , n) are positive constants.

Problem 3. Find the solution of the equation

e−x − 1 = 2(√

2 + 1).

Equations 99

Problem 4. Let n be a positive integer. Solve the nonlinear equation

1− ne−x

1 + ne−x=x

2

for n = 1, 2, 3, 4, 5.

Problem 5. Let c1 > 0 and c2 > 0. What is the condition on c1, c2 suchthat

x1 + x2 = c1, x1x2 = c2

has a real solution?

Problem 6. Let n be a positive integer. Solve

1− ne−x

1 + ne−x=x

2

for n = 1, 2, 3, 4, 5.

Problem 7. Let x > 0. Show that solution of the equation

x(1 + x1/2) = (1− x1/2)

is given by

x =19

((17 + 3√

33)1/3 + (17− 3√

33)1/3 − 1)2 ≈ 0.2955977 . . .

Problem 8. Let x1, x2, x3 ∈ R. Find solutions of the equation

x21x

22 + x2

2x23 + x2

3x21 = x1x2x3

with x1 6= 0, x2 6= 0, x3 6= 0. Note that

x1 = x2 = x3 = 1/3, x1 = 1/3, x2 = x3 = −1/3, x1 = x2 = −1/3, x3 = 1/3, x1 = x3 = −1/3, x2 = 1/3

are solutions.

Problem 9. (i) Consider the function

f(z, z) = z + z + zz.

Find all solutions of f(z, z) = 0.(ii) Let A be an n× n matrix over C. Find all solutions of

A+A∗ +AA∗ = 0n.


Find all 2× 2 matrices A such that

A+A∗ +AA∗ = I2.

Problem 10. (i) Let φ, θ ∈ [0, 2π). Consider the equation

eiφ sin(θ) =2η

|η|2 + 1.

Let η = 0, 1,−1, i,−i. Find φ and θ.(ii) Let θ ∈ [0, 2π). Consider the equation

cos(θ) =|η|2 − 1|η|2 + 1

.

Let η = 0, 1,−1, i,−i. Find φ and θ.

Problem 11. Show that roots of the polynomial f : R→ R

f(x) = x4 − x3 − 10x2 − x+ 1

are given by

x1 =12

(√

5− 3), x2 =12

(−√

5− 3), x3 = 2 +√

3, x4 = 2−√

3.

Problem 12. Find the real root of the polynomial f : R→ R

f(x) = x3 + x− 1.

The real root lie in the interval [0, 1]. Note that f(0) = −1 and f(1) = 1.

Problem 13. Find solution solutions of the equation (Scherk’s surface)

cos(x2)ex3 = cos(x1).

For example (x1, x2, x3) = (0, 0, 0) is a solution and (x1, x2, x3) = (π/3, π/3, 0)is a solution.

Problem 14. Let x > 0. Find all solutions of

x =√

1 +√

2 + x.

First show that x4 − 2x2 − x− 1 = 0.

Equations 101

Problem 15. Show that the system of equations

3x+ y − z + u2 = 0x− y + 2z + u= 0

2x+ 2y − 3z − 2u= 0

can be solved for x, y, u in terms of z; for x, z, u in terms of y; for y, z, uin terms of x; but not for x, y, z in terms of u.

Problem 16. Are there solutions of the equation (z ∈ C)

sin(z) = ez and cos(z) = ez ?

Problem 17. (i) Show that cos(π/7) is a real root of

8x4 + 4x3 − 8x2 − 3x+ 1 = 0.

(ii) Show that

cos(π/7) cos(2π/7) cos(4π/7) = −1/8.

Problem 18. Draw the curve in the plane given by

sinh(x) = exp(−2y).

Note that if y = 0, then sinh(2x) = 1 and therefore 2x = arcsinh(1).

Problem 19. Solve the system of integral equations∫ 2π

0

(f(φ))3 cos(φ)dφ = 0,∫ 2π

0

(f(φ))3 sin(φ)dφ = 0.

Problem 20. Consider the two circles in the plane

(x1 + 4)2 + (x2 + 5)2 = 194, (x1 − 3)2 + (x2 − 2)2 = 40.

Show that the two circles intersect and find the points of intersection. Showthat x1 + x2 = 9.

Chapter 6

Normed Spaces

Let C be the field of complex numbers. Let V be a vector space (linearspace) over C. Then V is a normed linear space if for every f ∈ V there isa real number ‖f‖ such that (c ∈ C)

(i) ‖f‖ ≥ 0

(ii) ‖f‖ = 0 if and only if f = 0

(iii) ‖cf‖ = |c| ‖f‖for every c ∈ C

(iv) ‖f + g‖ ≤ ‖f‖+ ‖g‖.

A normed linear space V which does have the property that all Cauchysequences are convergent is said to be complete. A complete normed linearspace is called a Banach space.

102

Normed Spaces 103

6.1 Solved Problems

Problem 1. Consider the Banach space R4 and the four vectors

v1 =

1x

x2/2x3/3

, v2 =

01xx2

, v3 =

001

2x

. v4 =

0002

where x ∈ R. Show that the vectors are linearly independent. Find thedistances

‖v1 − v2‖, ‖v2 − v3‖, ‖v3 − v4‖, ‖v4 − v1‖.

Problem 2. Consider the Hilbert space C2 and the vectors

v1 =(

cos(i)sin(i)

), v2 =

(− sin(i)cos(i)

).

Find the distance ‖v1 − v2‖.

Problem 3. (i) Let a, b ∈ R. Show that

d(a, b) := | arctan(a)− arctan(b)|

defines a distance in R.(ii) Show that xn = arctan(n), (n ∈ N) is a Cauchy sequence. Is the metricspace R , d complete?

Problem 4. Let n ≥ 1, 0 ≤ a < b and p ∈ Rn. Show that there existsa map k : C∞(Rn,R) such that k(x) = 0 for ‖x − p‖ ≥ b, k(x) = 1 for‖x− p‖ ≤ a, and 0 < k(x) ≤ 1 for ‖x− p‖ ≤ b.

Problem 5. Let x ∈ R2. Is

‖x‖ := |x1x2|1/2

a norm on R2?

Problem 6. Show that if a function f : R→ R satisfies

|f(x)− f(y)| ≤M(|x− y|)a

for some fixed M > 0 and a > 1, then f is a constant function, i.e., f isidentically equal to some real number b for all x ∈ R.


Problem 7. Let f , g be continuously differentiable functions on theinterval [0, 1]. One defines

〈f, g〉 =∫ 1

0

(f(x)g(x) +

df

dx

dg

dx

)dx.

Show that this satisfies the properties of an inner product. Calculate 〈f, g〉for f(x) = sin(x), g(x) = cos(x). Extend it to

〈f, g〉 =∫ 1

0

n∑j=0

djf(x)dxj

djg(x)dxj

dx.

Problem 8. The p-norm of a vector (x1, . . . , xn) ∈ Rn is defined as

‖x‖p :=

n∑j=1

|xj |p1/p

where p ∈ R+. Find the norm for p→∞.

Problem 9. Let x ∈ Rn and ‖x‖ be the Euclidean norm of x. If B ⊂ Rmis a nonempty compect set and x ∈ Rn, then we define the distance

dist(x, B) := min ‖x− b‖ : b ∈ B .

If A,B ⊂ Rn are nonempty compact sets then we define the distance

dist(A,B) := max dist(a, B) : a ∈ A .

Show thatdist(A,B) = 0 ⇔ A ⊂ B

and dist(A,B) < ε means that A ⊂ Nε(B) where

Nε(B) := x : dist(x, B) < ε

is the ε-neighborhood of B.

Problem 10. Let v and w be two normalized column vectors in Cn.Does

D(v,w) := 2 arccos

(√(v∗w)(w∗v)(v∗v)(w∗w)

)provide a distance measure between v and w.

Normed Spaces 105

Problem 11. Let z1, z2 ∈ C. One can define a distance measure via

ρ(z1, z2) =|z1 − z2|√

(1 + |z1|2)(1 + |z2|2)

Let z1 = eiφ1 and z2 = eiφ2 . Find ρ(z1, z2).

Problem 12. The chordal distance between to complex numbers z1, z2

is defined as

d(z1, z2) :=|z1 − z2|√

1 + |z1|2√

1 + |z2|2.

Let z1 = eiπ/2 and z2 = e−iπ/2. Find d(z1, z2).

Problem 13. Find n× n matrices A, B such that

‖[A,B]− In‖ → min

where ‖.‖ denotes the norm and [, ] denotes the commutator.



Problem 1. Let x, y ∈ R. Is

d(x, y) =|x− y|

2 + |x− y|

a metric on R? Note that d(x, x) = 0 and d(x, y) ≥ 0.

Problem 2. Let R be the set of real numbers and x, y ∈ R. Show that

d(x, y) = |x− y|

provides a distance function. Let x = (x1, . . . , xn) ∈ Rn and y = (y1, . . . , yn) ∈Rn Show that

d(x,y) =

√√√√√ n∑j=1

(xj − yj)2

provides a distance function. Of course one always assume the positivesquare root.

Problem 3. Consider the functions f : R→ R, g : R→ R

f(x) = exp(−x2), g(x) = exp(−|x|).

Find the distance d(f, g) between f and g given by

d(f, g) =∫

R

|f(x)− g(x)|1 + |f(x)− g(x)|

dx.

Chapter 7

Complex Numbers andComplex Functions

7.1 Solved ProblemsProblem 1. Let z ∈ C. Solve the nonlinear equation

z3 = z|z|2.

Problem 2. (i) Find the complex numbers z satisfying z2 = z.(ii) Find the complex numbers z satisfying z3 = z.

Problem 3. Solve the cubic equation z3 = −1. Do the solutions forma group under multiplication? If not, what numbers have to be added toform a group. Find them by multiplication of the solutions of the cubicequation.

Problem 4. Find the square root of z = 4 exp(iπ/4).

Problem 5. Let φ ∈ [0, 2π). Consider the complex number z = 1− eiφ.Find the condition on φ such that |z| = 1.

107


Problem 6. Let x, y ∈ R and z = x + iy. Find the real and imaginarypart of i/(πz).

Problem 7. Let φ ∈ [0, 2π) and θ ∈ (−π, π). Can any complex numberbe represented by

z = eiφ tan(

12θ

)?

Problem 8. Find

zdz − zdz, (zdz − zdz)⊗ (zdz − zdz).

Problem 9. Let x, c ∈ R and c 6= 0. Find the real and imaginary part ofthe function

fc(x) =x− icx+ ic

.

Problem 10. Let x, y ∈ R. Solve

x− ix+ i

y − iy + i

=xy − ixy + i

.

Problem 11. Find√i.

Problem 12. (i) Calculate

p(α, β, θ, φ) = | cos(α) cos(β) sin(θ)eiφ + sin(α) sin(β) cos(θ)|2.

(ii) Show that p(α, β, θ, φ) ≤ 1.(iii) Simplify the result from (i) for θ = π/4 and φ = 0.

Problem 13. Let z1, z2 ∈ C. Consider the distance measures

d1(z1, z2) := |z1 − z2|, d2(z1, z2) :=|z1 − z2|

1 + |z1 − z2|,

d3(z1, z2) :=|z1 − z2|√

(1 + |z1|2)(1 + |z2|2).

Here d3 is the chordal distance. Let z1 = eiφ and z2 = eiφ2 . Find d1, d2,d3.

Complex Numbers and Complex Functions 109

Problem 14. Let f(z1) and g(z2) be a pair of analytic functions of z1

and z2, respectively. We define

f(z1) g(z2) :=1

2π

∫ 2π

0

f(z1eiθ)g(z2e

−iθ)dθ.

Let

f(z1) =∞∑k=0

akzk1 , g(z2) =

∞∑k=0

bkzk2

for |zj | < R, j = 1, 2. Find f(z1) g(z2).

Problem 15. Let z = x+ iy, where x, y ∈ R. Find

∂

∂z,

∂

∂z,

∂

∂x,

∂

∂y.

Problem 16. Study the behaviour (stability) of the fixed points of thecomplex map f : C→ C, f(z) = z2.

Problem 17. Let z = x + iy (x, y ∈ R) be a nonzero complex number.We define the principal argument by z = |z| exp(iarg(z)), where arg(z) ∈(−π, π] and we define the imaginary remainder Imr(z) and the imaginaryquotient Imq(z) by

=(z) = Imr(z) + 2πImq(z)

where Imr(z) ∈ (−π, π] and Imq(z) ∈ Z. Show that

ln(ez) = <(z) + iImr(z)

and in particular

ln(ez) = z iff =(z) ∈ (−π, π].

Problem 18. Let z ∈ C and consider the analytic map

f(z) = exp(z).

Find the solutions of the equation

z = f(z).

This are the fixed points of f . We set z = x+ iy (x, y ∈ R. Then

x+ iy = exp(x+ iy) ≡ exeiy = ex(cos(y) + i sin(y)).


Thus we have to solve

ex cos(y)− x = 0, ex sin(y)− y = 0.

Problem 19. Let A be an n×n matrix. Suppose f is an analytic functioninside on a closed contour Γ which encircles λ(A), where λ(A) denotes theeigenvalues of A. We define f(A) to be the n× n matrix

f(A) =1

2πi

∮Γ

f(z)(zIn −A)−1dz.

This is a matrix version of the Cauchy integral theorem. The integral isdefined on an element-by-element basis f(A) = (fjk), where

fjk =1

2πi

∮Γ

f(z)eTj (zIn −A)−1edz.

Let f(z) = z2 and

A =(

0 11 0

).

Calculate f(A).

Problem 20. Let x1, y1, x2, y2 ∈ R. Consider the two complex numbers

z1 = x1 + iy1, z2 = x2 + iy2.

In polar form we have

z1 = r1eiφ1 = r1(cos(φ1)+ i sin(φ1)), z2 = r2e

iφ2 = r2(cos(φ2)+ i sin(φ2))

where r21 = x2

1 + y21 , r2

2 = x22 + y2

2 . Now

z = z1 + z2 = (x1 + x2) + i(y1 + y2).

Find z in polar form z = reiφ.

Problem 21. Let a, b, φ be fixed real numbers. Consider the function

w(z) = z + az2 + beiφz3.

Write it as real and imginary part, with w = u+ iv and z = x+ iy.

Problem 22. Consider the z-transform

x(z) =∞∑n=0

x(n)z−n, x(n) =1

2πi

∮x(z)zn−1dz.


Let

S(N) :=N∑n=1

x(n).

Then

S(N) =N∑n=1

x(n) =1

2πi

∮x(z)

N∑n=1

zn−1dz.

It follows that (geometric series)

S(N) =1

2πi

∮x(z)(zN − 1)

z − 1dz.

Apply this expression and the residue theorem to calculate

S(N) =N∑n=1

n3.

Problem 23. (i) Consider the complex number z = eiφ. Let n ∈ N. Findn√z.

(ii) Let z = reiφ and w = x+ iy (x, y ∈ R). Find zw.

Problem 24. Consider the complex numbers z1 = 0.4 + 0.3i, z2 = 5 + 2i.Calculate zz21 . Hint: Set z1 = r1e

iφ1 .

Problem 25. Consider the complex numbers

z1 = x1 + iy1 = 1 + 4i, z2 = x2 + iy2 = 3− 2i.

Calculate logz2(z1).

Problem 26. Let A > 0 and B ≥ 0. Consider the quadratic conformalmap in the complex w-plane of the unit disc in the complex z-plane

w(z) = Az +Bz2.

Let θ be the polar angle in the complex z-plane.(i) Show that with the notation w := u + iv, z := x + iy (with u, v, x, yreal) one has the parametric equation of the boundary

u(θ) = A cos(θ) +B cos(2θ), v(θ) = A sin(θ) +B sin(2θ).

(ii) Let C := B/A. Show that for C = 0 (i.e. B = 0) one obtains a circulardisc. Show that for C = 1/4 the curvature vanishes at θ = π. Show thatfor C = 1/2 the derivative dw/dz vanishes at the boundary, i.e. at θ = π.


Problem 27. (i) Solve the equation(z + i

2

z − i2

)4

= 1.

(ii) Solve the system of equations(z1 + i

2

z1 − i2

)4

=z1 − z2 + i

z1 − z2 − i,

(z2 + i

2

z2 − i2

)4

=z2 − z1 + i

z2 − z1 − i.

These equations play a role for the Bethe ansatz for spin systems.

Problem 28. Let z be a complex number such that |z| <∞ and <(z) > 0.Consider

Z(z) =∫ ∞

0

exp(−zx− x2)dx.

Show that

Z(z) =12

∞∑k=0

(−1)kΓ( 1

2 + 12k)

k!zk.

Problem 29. Let z 6= 0. Show that the function

f(z) =ln(z)z − 1

is analytic near z = 1 and admits the Taylor expansion

f(z) =∞∑n=0

(−1)n

n+ 1(z − 1)n for |z − 1| < 1.

Problem 30. Is the function f : C→ C

f(z) = z + |z|

continuous? Find the fixed points of f .

Problem 31. Let x1, x2, y1, y2 ∈ R and z1 = x1 + iy1, z2 = x2 + iy2 withz1z1 6= 0, z2z2 6= 0. Find the conditions such that

( z1 z2 )(z2

−z1

)= 0.


Problem 32. Let z = ρeiθ and ζ = Reiω with with R > 0 and ρ > 0.Show that

ρ2 −R2

ρ2 − 2Rρ cos(ω − θ) +R2= <

(z + ζ

z − ζ

).



15

(−1 + 2i) =1√5ei(π−arctan(2)).

Problem 2. Let z1 = r1eiφ1 , z2 = r2e

iφ2 with r1, r2 ≥ 0. Show that

<(z1 + z2

z1 − z2

)=

r21 + r2

2

|z1 − z2|.

Problem 3. Let z = x+ iy with x, y ∈ R. Show that

<(z2) = x2 − y2.

Problem 4. Let 0 < r < 1 and φ ∈ [0, 2π). Consider the map

f(z) = reiφ + (1− reiφ)z.

Find the fixed points. Find f(0), f(f(0)), f(f(f(0))).

Problem 5. Let x, y ∈ R. Find real and imaginary part of

z =

√x− iyx+ iy

eiφ.

Problem 6. Let z ∈ C. Show that

sin(z) = z

∞∏k=1

(1− z2

k2π2

).


Problem 7. Show that the Bessel function

Jν(z) =∞∑n=0

(−1)n

n!Γ(ν + n+ 1)

(z2

)2n+ν

is an entire function of z for ν = 0, 1, . . .. Show that

Jν−1(z) + Jν+1(z) = 2ν1zJν(z).

Problem 8. (i) Show that the radius of convergence of the function

f(z1, z2) =∞∑k=0

zk1z2

is given by (z1, z2) : |z1| < 1 ∨ z2 = 0 .(ii) Show that the radius of convergence of the function

f(z1, z2) =∞∑k=0

(z1z2)k

is given by (z1, z2) : |z1| · |z2| < 1 .(iii) Show that the radius of convergence of the function

f(z1, z2) =∞∑k1=0

k1∑k2=0

(k1

k2

)zk21 zk1−k22

is given by (z1, z2) : |z1|+ |z2| < 1 .

Problem 9. (i) Show that the sum

∞∑k=0

k!zk

only converges for z = 0.(ii) Show that the sum

∞∑k=0

zk

k2

converges for |z| ≤ 1 and diverges for |z| > 1.(iii) Show that the sum

∞∑k=0

(z1z2)k


converges for (z1, z2) : |z1| · |z2| < 1 .(iv) Show that the sum

∞∑k1=0

k1∑k2=0

(k1

k2

)zk21 zk1−k22

converges for (z1, z2) : |z1|+|z2| < 1 . The sum appears at the expansionof 1/(1− (z1 + z2)).

Problem 10. Let x ∈ R+ and f(x) =√x. Show that real function

can be extended with a power series expansion around 1 into the complexdomain

f(z) =√

1 + (z − 1) =∞∑k=0

(1/2k

)(z − 1)k

and for z = reiφ with −π/2 < φ < π/2 we have f(z) =√reiφ/2.

Problem 11. (i) Is the complex function

f(z) = z = x1 − ix2

holomorph?(ii) Is the complex function

f(z) = z2 − z2 = 4ix1x2

holomorph?

Problem 12. (i) Show that if n is a positive integer then

(r(sin(φ) + i sin(φ)))n ≡ rn(cos(nφ) + i sin(nφ)). (1)

Hint. Apply exp(iφ) ≡ cos(φ) + i sin(φ).(ii) Show that the n-th roots of unity are

ρ = cos(

2πn

)+ i sin

(2πn

), ρ2, ρ3, . . . , ρn−1, ρn ≡ 1. (1)

Problem 13. Let ε ∈ R and z ∈ C. Consider the product

f(z) =∞∏k=1

1− e−εk

1− e−ε(k+z).


Find f(0). Show that

f(z) = (1− e−εz)f(z − 1).

Show that

f(m) =m∏k=1

(1− e−εk), m = 1, 2, . . .

Show that f is periodic with period 2πi/ε. Show that f has simple polesat z = −m (m = 1, 2, . . .). Show that the residues are given by

limz→−m

(z +m)f(z) =(−1)m+1 exp(− 1

2εm2 + 1

2εm)εf(m− 1)

.


(1 + i)1/2 = ±21/4 (cos(π/8) + i sin(π/8)) .

Problem 15. Let n = 0, 1, . . . and x ∈ R. Find the real and imaginarypart of the functions

fn(x) =1√π

(ix− 1)n

(ix+ 1)n+1.


1(n− k)!

=1

2πi

∮dt

et

tn−k+1

where the integration contour is a small circle around the origin in thecomplex plane.

Problem 17. We know that 0 ≤ | sin(x)| ≤ 1 and 0 ≤ | cos(x)| ≤ 1 forx ∈ R. This is no longer true for sin(z) and cos(z) with z ∈ C.(i) Let a > 0. Show that

| sin(az)| =√

sinh2(ay) + sin2(ax), | cos(az)| =√

sinh2(ay) + cos2(ax).

(ii) Find all solutions of | sin(z)| = 2 and | cos(z)| = 2.(iii) Find all solutions of cos(z) = i.


(iv) Let n ∈ N. Show that (x, y ∈ R)

| sin(n(x+ iy))|=√

sin2(nx) cosh2(ny) + cos2(nx) sinh2(ny)

> | sinh2(ny)| → ∞

as n→∞ for any y 6= 0.

Chapter 8

Integration

8.1 Solved ProblemsProblem 1. The time-average of a continuous function f is

〈f〉 := limT→∞

12T

∫ T

−Tf(t)dt.

Find the time-average of the functions

f1(t) = cos(ωt) sin(ωt), f2(t) = cos2(ωt), f3(t) = sin2(ωt).

Problem 2. Let j, k = 1, 2, . . .. Consider the function

f(j, k) =∫ 1

0

xj−1(1− x)k−1dx, j, k = 1, 2, . . . .

Thus f(1, 1) = 1. Is

f(j − 1, k + 1) =k

j − 1f(j, k), j ≥ 2

andf(j + 1, k) = f(j, k)− f(j, k + 1) ?

Prove or disprove.

118

Integration 119

Problem 3. Let

x(τ) =

1 for 0 ≤ τ ≤ 10 otherwise

and

h(τ) =

1 for 0 ≤ τ ≤ 10 otherwise .

Find the convolution integral

y(t) =∫ ∞−∞

x(τ)h(t− τ)dτ.

Problem 4. Let T > 0 and ω = 2π/T . Let m,n ∈ N and α, β ∈ R.Calculate

I(α, β) =1T

∫ T

0

cm sin(mωt+ φm − α)cn sin(nωt+ φn − β)dt.

Problem 5. A cubic B-spline with uniform knot spacing, centered at theorign, is given by

B(x) =

16 (2− |x|)3 if 1 ≤ |x| < 2

16 ((2− |x|)3 − 4(1− |x|)3) if 0 ≤ |x| < 1

0 otherwise

Find the integral ∫ ∞−∞

B(x)dx.

Problem 6. Find the normalization constant K from the condition

1 = 4K∫ 2π

0

sin2(ω/2)dω∫ π

0

sin(θ)dθ∫ π

−πdφ.

Problem 7. Let ε ∈ R. Calculate the integral

f(ε) =∫ 2π

0

exp(εeiφ)dφ

by finding an ordinary differential equation for f together with the initialconditions. Obviously f(0) = 2π.


Problem 8. Let b > a. Find the mean and variance of random variablex with uniform probabilty density function p

p(x) =

1b−a if a ≤ x ≤ b0 otherwise

Problem 9. Consider a one-dimensional lattice (chain) with lattice con-stant a. Let k be the sum over the first Brioullin zone we have

1N

∑k∈1.BZ

F (ε(k))→ a

2π

∫ π/a

−π/aF (ε(k))dk = G

whereε(k) = ε0 − 2ε1 cos(ka).

Using the identity ∫ ∞−∞

δ(E − ε(k))F (E)dE ≡ F (ε(k))

we can write

G =a

2π

∫ ∞−∞

F (E)

(∫ π/a

−π/aδ(E − ε(k))dk

)dE.

Calculate

g(E) =∫ π/a

−π/aδ(E − ε(k))dk

where g(E) is called the density of states.

Problem 10. Let

i1(t) = I2 sin2(ωt), i2(t) = I2 sin2(ωt+ φ).

Calculate

〈i1(t)i2(t)〉 := limT→∞

1T

∫ T

0

i1(t)i2(t)dt

and〈i1(t)i2(t)〉 − 〈i1(t)〉〈i2(t)〉.

Problem 11. Let α > 0. Find∫ ∞0

exp(−αx)dx =1α.

Integration 121

Problem 12. Let f(t) be a continuous function. Show that∫ x

0

dζ

∫ ζ

0

f(t)dt =∫ x

0

(x− t)f(t)dt.

Problem 13. (i) Calculate∫ ∞−∞

sech(t)tanh(t) cos(t+ t0)dt.

(ii) Calculate ∫ ∞−∞

sech2(t)tanh2(t)dt.

Problem 14. Let λ > 0. Calculate∫ ∞−∞

sin(λt)λ sinh(t)

dt.

Problem 15. Calculate∫ ∞−∞

exp(−x2 + 2ixy)dx.

Problem 16. Calculate the integral

I(λ) =∫ ∞−∞

et

(1 + et)2cos(t+ λ)dt

using the residue technique.

Problem 17. Let m be an non-negative integer. Find∫ π

0

dθ| cos2m+1(θ)|.

Problem 18. Let ε > 0. Calculate∫ ∞0

k2dk

eεk2 − 1.


Problem 19. The mother Haar wavelet is given by

f(t) =

−1 for 0 ≤ t < 1/2+1 for 1/2 ≤ t < 10 otherwise

Find the Fourier transform

f(ω) =∫ ∞−∞

e−iωtf(t)dt.

Problem 20. The Poisson wavelet is given by

f(t) =(td

dt+ 1)P (t)

whereP (t) =

1π

11 + t2

.

Find the Fourier transform of f .

Problem 21. Let m ∈ Z. Calculate

am =1

2π

∫ 2π

0

eimφ2 cos(φ)dφ.

Problem 22. Let 0 < α < 1. Find the integral∫ ∞0

xα−1

1 + xdx.

Problem 23. The linear one-dimensional diffusion equation is given by

∂u

∂t= D

∂2u

∂x2, t ≥ 0, −∞ < x <∞

where u(x, t) denotes the concentration at time t and position x ∈ R. Dis the diffusion constant which is assumed to be independent of x and t.Given the initial condition c(x, 0) = f(x), x ∈ R the solution of the one-dimensional diffusion equation is given by

u(x, t) =∫ ∞−∞

G(x, t|x′, 0)f(x′)dx′

Integration 123

where

G(x, t|x′, t′) =1√

4πD(t− t′)exp

(− (x− x′)2

4D(t− t′)

).

HereG(x, t|x′, t′) is called the fundamental solution of the diffusion equationobtained for the initial data δ(x− x′) at t = t′, where δ denotes the Diracdelta function.(i) Let u(x, 0) = f(x) = exp(−x2/(2σ)). Find u(x, t).(ii) Let u(x, 0) = f(x) = exp(−|x|/σ). Find u(x, t).

Problem 24. Let ω0 > 0 be a fixed frequency and t the time. Calculate

f(ω) =1√2π

∫ +∞

−∞e−|ωt|e−iωtdt.

Problem 25. Let ω0 > 0 be a fixed frequency and t the time. Calculate

f(ω) =1√2π

∫ ∞−∞

e−|ω0t|e−iωtdt.

Problem 26. The sum

limn→∞

1n

n∑k=1

exp(

2 cos(

kπ

n+ 1

))can be cast into the integral

limn→∞

1n

∫ n

0

exp(2 cos(πx))dx. (1)

Calculate this integral.

Problem 27. Let f : R → R be an analytic function. The Dirichletintegral identity is given by∫ u

0

∫ u−w2

0

f(u− w1 − w2)wµ1−11 wµ2−1

2 dw1dw2 =

Γ(µ1)Γ(µ2)Γ(µ1 + µ2)

∫ u

0

f(u− w)wµ1+µ2−1dw.

Let f(x) = e−x. Calculate the left and right-hand side of this identity.


Problem 28. Let r1 > 0, r2 > 0. Find the integral

I(r1, r2) =∫ 1

−1

dx√r21 + r2

2 − 2r1r2x.


ea2

=∫

Rdx exp

(−x

2

2+√

2ax).

Problem 30. Let u ≥ 0 and

ρ(u) =12

exp(−√u).

Find

ρn =∫ −∞

0

unρ(u)du.


I =∫ ∫

D

√x2 + 4y2dydx

where D is the domain bounded by the positive x-axis, the positive y-axisand the parabola y2 = 1− x.

Problem 32. Calculate the definite integral∫ 1

0

sin(x2)dx.

Problem 33. (i) Consider the wavelet (ω0 > 0)

ψ0(t) = (eiω0t − e−ω20/2)e−t

2/2.

Show that ∫ ∞−∞

ψ0(t)dt = 0.

Hint: Use (a > 0)∫ ∞−∞

e−(ax2+bx+c)dx =√π

ae(b2−4ac)/4a.

Integration 125

(ii) We define

ψn(t) :=d

dtψn−1(t), n = 1, 2, . . .

Show that ∫ ∞−∞

tkψn(t)dt = 0, 0 ≤ k ≤ n.

Problem 34. Let b > a. Consider the integral∫ y=b

y=a

f(y)dy

where f is a continuous function in [a, b]. Apply the transformation

y(x) =12

((b− a)x+ a+ b)

so that the integration range is between −1 and +1. Then the Gaussquadrature can be applied which extends over the interval [−1,+1].

Problem 35. Let ε > 0. Find f of the equation

exp(−εt) = 1−∫ t

0

f(s)ds.

Problem 36. (i) Find the area of the set

S2 := (x1, x2) : 1 ≥ x1 ≥ x2 ≥ 0 .

(ii) Find the volume of the set

S3 := (x1, x2, x3) : 1 ≥ x1 ≥ x2 ≥ x3 ≥ 0 .

Extend the n-dimensions.

Problem 37. Let ω1, ω2 be real and positive. Find

J(ω1, ω2) =1√2π

∫ ∞−∞

exp(−12x2 + iω1x+ iω2x

2)dx.

Problem 38. Let e be the eccentricity of an ellipse, i.e.√

1− e2 = b/a.Let n ≥ 1. Calculate the integral∫ 2π

0

dx(1e − cos(x)

)n


by making the substitution z = exp(ix). Apply the residue theorem.Hint. We have∫ 2π

0

dx(1e − cos(x)

)n =∫|z|=1

dz(1e −

12

(z + 1

z

))niz

=∫|z|=1

i(−1)n+12nzn−1dz(z2 − 2

ez + 1)n .

Problem 39. Let q2 > 0 and (p− q)2 > 0. Find∫ ∞0

exp(−xq2)dx,∫ ∞

0

exp(−x(p− q)2)dx.

Problem 40. Find α > 0 such that∫ +∞

−∞exp(−α|x|)dx = 1.

Afterwards calculate∫ ∞−∞

x exp(−α|x|)dx,∫ ∞−∞

x2 exp(−α|x|)dx.

Problem 41. Let c > 0. Show that

exp(

12cy2

)=

1√2π

∫ ∞−∞

dx exp(−1

2x2 +

√cyx

).

Problem 42. The Gauss invariant for two given closed loops Cα and Cβin R3 parametrized by rα(s), rβ is defined by

G(Cα, Cβ) :=1

4π

∮Cα

ds

∮Cβ

ds′drα(s)ds

× drβ(s′)ds′

rα(s)− rβ(s′)|rα(s)− rβ(s′)|3

where × denotes the vector product. Find G(Cα, Cβ) for the two curves

Cα : x21 + x2

3 = 1, Cβ : (x1 − 1)2 + x22 = 1.

Problem 43. Let x ∈ R. We define [x] as the integer part of x andx := x− [x]. Calculate the integrals∫ 7/2

0

[x]dx,∫ 7/2

0

xdx.

Integration 127

Problem 44. Let x ∈ R. Consider the integral

f(x) =∫ ∞x

eiy

ydy.

(i) Show that f(−|x|) = f(|x|)− iπ.(ii) Show that for large x

f(x) = eix(i

x+

1x2

+ · · ·).

Problem 45. Let n = 0, 1, 2, . . .. We define

an :=∫ π/2

0

cos2n(x)dx, bn :=∫ π/2

0

x2 cos2n(x)dx.

Then a0 = π/2 and b0 = π3/24. Show that using integration by parts

an = (2n− 1)(an−1 − an).

Show that for n ≥ 1 we have

an = (2n− 1)nbn−1 − 2n2bn.

Show that (n ≥ 1)

0 ≤ π2

6−

n∑k=1

1k2

= 2bnan≤ π2

4(n+ 1).

Problem 46. Calculate

I =∫ 10

0

(x− int(x))dx

where int(x) defines the largest integer less than x.

Problem 47. Let 0 < a < b < 1. Find∫ b

a

ln(1− x)x

dx.

Problem 48. Show that∫ π/2

0

cos10(x) cos8(2x) cos6(4x) cos4(6x) cos2(8x)dx =


5166673π536870912

+296654976251281698120709987525225

.

Problem 49. Let z ∈ C. Let ln(1 + z) be the branch of the logarithmdefined on C \ (−∞,−1]. Calculate

In(r) = Pv

∫|z|=r

zn−1 ln(1 + z)dz, r > 0, n ∈ Z

where Pv is the principal value.

Problem 50. Let b > a and a, b be finite. Consider the integral∫ b

a

f(x)dx.

(i) Apply the transformation

x =12

(a+ b+ (b− a) tanh(y)⇔ y = tanh−1

(2x− a− bb− a

)to the integral.(ii) Apply the transformation

x =12

(a+ b+ (b− a) tanh

(π2

sinh(y)))

withdx

dy=

(b− a)π cosh(y)/4cosh2(π sinh(y)/2)

to the integral.

Problem 51. Calculate the integral∫ π

0

1 + sin(x)1 + cos(x)

dx

utilizing the transformation t = tan(x/2) with −π < x < π. From thistransformation it follows that

x = 2 arctan(t), dx =2

1 + t2dt

and x = 0→ t = 0, x = π/2→ t = 1.

Integration 129

Problem 52. Simplify the integral

I =∫ 1

0

cos(x)√x

dx

for numerical calculation.

Problem 53. Let a > 0. Let f : R → R, g : R → R be continuousfunctions with f(−x) = f(x) and g(−x) = −g(x). Show that∫ a

−af(x)g(x)dx = 0.

Problem 54. Show that∫ ∞0

e−xx2dx = 2!,∫ ∞

0

e−xx3dx = 3!.

Let n = 4, 5, . . .. Show that∫ ∞0

e−xxndx = n!.

Problem 55. Let k1 > 0, k2 > 0, k3 > 0. Find the integral∫ ∞0

sin(k1r) sin(k2r) sin(k3r)dr

r.

Problem 56. Calculate

F (s, t) =∫

Re−ixt−|x−s|dx.

Problem 57. Let x ∈ (0, 1). Calculate

P

∫ 1

0

ym

(y − x)dy

for m = 0, m = 1, m = 2, m = 3.

Problem 58. Let a, b > 0 and b > |a|. Show that∫R

dx

x2 + 2ax+ b2=

π√b2 − a2

.


Problem 59. Let a, b > 0. Show that∫ ∞0

e−ax − e−bx

xdx = ln

(b

a

).

Problem 60. Let x > 0. Show that∫ ∞0

e−txdt =1x.

Problem 61. a, b ∈ R. Let

f(x) = a exp(bx).

Find the conditions on a and b such that∫ 1

0

f(x)dx = 1,∫ 1

0

xf(x)dx =12.

Problem 62. Dawson’s integral is given by

f(x) =∫ x

0

et2−x2

dt, x ≥ 0.

(i) Show that for all complex values z the function f satisfies the lineardifferential equation

df(z)dz

+ zf(z) = 1.

(ii) Let j = 1, 2, . . .. Show that

f (j+1)(z) + 2zf (j) + 2jf (j−1)(z) = 0, j = 1, 2, . . .

where f (j) indicates the jth derivative.

Problem 63. Let a, b, c ∈ R and a+ b cos(θ) + c sin(θ) 6= 0. Show that∫ +π

−π

b sin(θ)− c cos(θ)a+ b cos(θ) + c sin(θ)

dθ = 0.

Problem 64. Let a > 0. Show that∫ ∞0

cos(ax)1 + x2

dx =π

2e−a.

Integration 131


ln(cosh(x))cosh3(x)

dx =π

4(ln(2)− 1/2).

Problem 66. Let m,n = 0, 1, 2, . . .. Find the integral

fmn(t) =∫ t

0

(t− τ)mτndτ.

Problem 67. Let a, b > 0. Find the integral∫ ∞0

cos(at)− cos(bt)t

dt.

Problem 68. Let q2 > 0. Calculate the integral∫ ∞0

exp(−q2x)dx.

Problem 69. Let x > 0. Assume that f : R→ R is integrable over [0, x]for all x > 0 and limx→∞ f(x) = a. Show that

limx→∞

1x

∫ x

0

f(s)ds = a.

Utilize∣∣∣∣ 1x∫ x

0

f(s)ds− a∣∣∣∣ =

∣∣∣∣ 1x∫ x

0

(f(s)− a)ds∣∣∣∣ ≤ ∣∣∣∣ 1x

∣∣∣∣ ∫ x

0

|f(s)− a|ds.

Problem 70. Consider the function f : [0, 1]→ [0, 1]

f(x) =

1/x− int(1/x) x 6= 00 x = 0

where int(y) denotes the integer part of y. The function is integrable.(i) Let k ≥ 1 be a positive integer. Find∫ 1/k

1/(k+1)

f(x)dx.


(ii) Find ∫ 1

1/k

f(x)dx.

(iii) Find ∫ 1

0

f(x)dx.

Problem 71. Show that∫ π/2

0

√sin(x)√

sin(x) +√

cos(x)dx =

π

4.

Problem 72. Calculate ∫ 3

0

(x− bxc+12

)dx

where bxc denotes the greatest integer less than or equal to x.

Hilbert Transform

Problem 73. The Hilbert transform H and its inverse is given by

g(y) =H(f(x)) =1πP

∫R

f(x)x− y

dx

f(x) =H−1(g(y)) =1πP

∫R

g(y)y − x

dy

where the Cauchy principal value is defined by

P

∫Rf(x)dx := lim

R→∞

∫ +R

−Rf(x)dx.

The Hilbert transform relates parts of the function in the same domain.Let k > 0. Find the Hilbert transform of f(x) = cos(kx).

Problem 74. Consider the Hilbert transform

H[f ] =1πP

∫ ∞−∞

f(x′)x′ − x

dx′.

Integration 133

Find H2.

Problem 75. The Hilbert transform acts on a one-dimensional functiong(s) by a convolution with the kernel 1/(πs). The singularity at s = 0 ishandled in the Cauchy principal value sense. The Fourier transform of theHilbert kernel is −i sgnσ. Thus the Hilbert transform of g is

Hg(s) =∫ ∞−∞

g(s− s′)πs′

ds′ =∫ ∞−∞

(−i sgnσ)G(σ)ei2πsσdσ

where G(σ) is the Fourier transform of g, i.e.

G(σ) =∫ ∞−∞

g(s)e−i2πσsds.

Suppose g is a function whose support is strictly less than radius R, i.e.g(s) = 0 for all |s| > R − ε for some small positive ε. Find an inversionformula.

Problem 76. The Hilbert transform of a function f ∈ L2(R) is definedas

H(f)(y) =1πPV

∫ ∞−∞

f(x)x− y

dx

where PV stands for principal value. Calculate the Hilbert transform off(x) = 1/(1 + x4).

Problem 77. The Radon transform for a function f(x, y) is given by theintegral transform

P (r, θ) = Rf(x, y) =∫ +∞

−∞f(r cos(θ)− s sin(θ), r sin(θ) + s cos(θ))ds.

The function P (r, θ) desribes the values of points on projections. Show thatthe inverse Radon transform can be given by

f(x, y) = R−1P (r, θ) = − 12πBHDP (r, θ)

where D is the partial differential operator Dg(r, θ) = ∂g/∂r with respectto r, H is the Hilbert transform operator

Hg(r, θ) = − 1π

∫g(u, θ)r − u

du

and B is the backprojection operator

Bg(r, θ) =∫ π

0

g(x cos(θ) + y sin(θ), θ)dθ.


Problem 78. The Hilbert transform of a function f ∈ L2(R) is definedas

H(f)(y) =1πPV

∫ ∞−∞

f(x)x− y

dx

where PV stands for principal value. Calculate the Hilbert transform of

f(x) = exp(−x2/2).

Problem 79. Let f : R→ R

f(x) =x

sinh(x).

Then f(0) = 1. Let c > 0. Find∫ c

0

f(x)dx.

Problem 80. Let p1, p2, p3 > 0. Calculate

f(p1, p2, p3) =4π

∫ ∞0

sin(p1r) sin(p2r) sin(p3r)dr

r.

Problem 81. Find the constant K (normalization) from the condition

1 = 4K∫ 2π

0

sin2(ω/2)dω∫ π

0

sin(θ)dθ∫ π

−πdφ.


I =1

8π2

∫ π

−πdα

∫ π

0

sin(β)dβ∫ π

−πF (α, β, γ)dγ.

(i) Find I for F (α, β, γ) = 1.(ii) Find I for F (α, β, γ) = α+ β + γ.(iii) Find I for F (α, β, γ) = αβγ.

Problem 83. Consider the circle around (0, 0, 0) in the x1 − x2 plane

r1(t) =

x1,1(t)x1,2(t)x1,3(t)

=

cos(t)sin(t)

0

, t ∈ [0, 2π]

Integration 135

and the circle around (0, 1, 0) in the x2 − x3 plane

r2(s) =

x2,1(s)x2,2(s)x2,3(s)

=

01 + cos(s)

sin(s)

.

Then the derivatives are

dr1(t)dt

=

− sin(t)cos(t)

0

,dr2(t)dt

=

0− sin(s)cos(s)

.

Calculate (Gauss formula)

14π

∮ ∮dtds

(dr1(t)dt

× dr2(s)ds

)· r1(t)− r2(s)|r1(t)− r2(s)|3

where × denotes the vector product, · denotes the scalar product and con-tour integrations run from 0 to 2π.


e−xx2dx = 2!,∫ ∞

0

e−xx3dx = 3!.

Problem 85. (i) Find the area of the set

S2 := (x1, x2) : 1 ≥ x1 ≥ x2 ≥ 0 .

(ii) Find the volume of the set

S3 := (x1, x2, x3) : 1 ≥ x1 ≥ x2 ≥ x3 ≥ 0 .

Extend the n-dimensions.

Problem 86. Let n be a positive integer. Find∫ π

0

cos2(nx)dx.

Problem 87. Let n ≥ 0. The Legendre polynomial of degree n is definedas

Pn(x) =1

2nn!dn

dxn(x2 − 1)


with P0(x) = 1. Let n ≥ 1. Show that∫ 1

0

xkPn(x)dx = 0

for k = 0, 1, . . . , n− 1

Problem 88. Let k = 0, 1, . . .. Calculate

ck =2π

∫ π

0

arcsin(cos(θ)) cos(kθ)dθ.

Problem 89. Find α > 0 such that∫ ∞0

2αxe−2αx3/3dx = 1.

Problem 90. The complete elliptic integral of first kind K(m) can bedefined by

K(m) =∫ 1

0

((1− t2)(1−mt2))−1/2dt, |m| < 1.

The beta function can be defined by

B(a, b) =∫ 1

0

ta−1(1− t)b−1dt, <(a) > 0, <(b) > 0.

Show that 2√

2K(1/2) = B(1/4, 1/2).

Problem 91. Consider the function f : [0, 1]→ [0, 1], f(x) =√x with

I =∫ 1

0

√xdx =

23x3/2

∣∣∣∣10

=23.

Find a approximation (upper bound) of the integral in the sense of Lebesgueby partioning the ordinate interval into the intervals [0, 1/4], (1/4, 1/2],(1/2, 3/4], (3/4, 1] with a0 = 0, a1 = 1/4, a2 = 1/2, a3 = 3/4, a4 = 1.

Problem 92. Let a > 0. Calculate∫dx

(a2 − x2)3/2.

Integration 137

Set x = a sin(θ). Then dx = a cos(θ)dθ.

Double Integrals

Problem 93. Let f : R2 → R be a continuous function and f(x1, x2) =f(x2, x1) for all x1, x2 ∈ R. Let b, a ∈ R and b > a. Calculate∫ b

a

∫ b

a

f(x1, x2) sin(x1 − x2)dx1dx2.

Problem 94. Let f be a continuous function.(i) Show that the double integral∫ x

0

dξ

∫ ξ

0

f(t)dt

can be expressed by a single integral.(ii) Show that (n ≥ 2)∫ x

0

dξ1

∫ ξ1

0

dξ2 · · ·∫ ξn−1

0

f(ξn)dξn

can be expressed by a single integral.


I =∫ π/2

0

sin2n(θ) cos2n+1(θ)dθ

by considering the double integral∫ ∫D

(r sin(θ))2n(r cos(θ)2n+1e−r2rdrdθ

where D is the first quadrant.

Problem 96. Find the integral

I(`) =∫ 2π

0

(∫ | cos(θ)|(`/2)

0

dp

)dθ.


Problem 97. Evaluate the quadruple integral

I =∫ 1

0

dx

∫ 1

0

dy

∫ 1

0

dx′∫ 1

0

dy′(1

((x− x′)2 + (y − y′)2)1/2− 1

((x− x′)2 + (y − y′)2 + 1)1/2

).

Problem 98. Let z ∈ (0, 1] and x ∈ (0, 1]. Find the double integral

f(z) = 1−∫ 1

z

dx

∫ 1

z/x

dy.

Problem 99. (i) Let r =√x2

1 + · · ·+ x2n. Show that the integral∫ ∫

r≥1

· · ·∫dx1 · · · dxn

rα

is finite for α > n and infinite for α ≤ n.(ii) Let r =

√x2

1 + · · ·+ x2n. Show that the integral∫ ∫

r≤1

· · ·∫dx1 · · · dxn

rα

is finite for α < n and infinite for α ≥ n.

Problem 100. Consider a three-dimensional probability distributionf(x1, x2, x3) such that for all j

fx(xj) =1

2√π

(1 + 2x2

j −−1 + 2x2

j√2

)e−x

2j

where fx(x) is the probability density associated with an individual vari-able. This means

fx(x1) =∫

R

∫Rf(x1, x2, x3)dx2dx3

etc. Is it possible that the probabilty density associated with the sum ofthese variables s = x1 + x2 + x3 is given by

fs(s) =1

2√π

(1 + 2s2 +

−1 + 2s2

√2

)e−s

2

Integration 139

provided that f(x1, x2, x3) is non-negative?

Problem 101. Consider a one-dimensional chain of length N with openend boundary conditions. The counting is from left to right starting at0. The canonical partition function Z(β) (β > 0) is given by the multipleintegral

ZN (β) =∫ 1

−1

ds0

∫ 1

−1

ds1 · · ·∫ 1

−1

dsN−1eβ|s0−s1|eβ|s1−s2| · · · eβ|sN−2−sN−1|.

Show that there is a coordinate transformation which decouples the sites.Find Z2(β) and Z3(β).

Problem 102. Let f : Rn → Rn be an analytic function. Consider themap

xj = f(xj−1) = · · · = f(x0).

To study the evolution of phase-space distributions, we can introduce theevolution operator U(x′,x, j) such that any initial phase-space distributionρ(x, 0) evolves into

ρ(x′; j) =∫

Ω

U(x′,x; j)ρ(x, 0)dx

where Ω is the phase space area. Find U(x′,x; j).

Problem 103. Let

S := (x, y) ∈ R2 : x, y ≥ 0, 0 ≤ x21 + x2

2 ≤ 1 .

Let m,n be nonnegative integers. Find the integral∫S

x2m+1y2n+1dxdy.

Problem 104. Calculate ∫ ∫A

dxdy

whereA = (x, y) : x, y ≥ 0, x+ y ≤ 1, x ≥ 1/3 .


I =∫ ∞

1

dx

∫ ∞1

dy

√x2 − 1

√y2 − 1

(x+ y)6


using the substitution x = cosh(α), y = cosh(β).

Problem 106. Find the integral

I =∫ ∞

1

dx

∫ ∞1

dy

√x2 − 1

√y2 − 1

(x+ y)6.

Hint. Use the substitutions x = cos(α), y = cosh(β) and show that theintegral can be written as

I =∫ ∞

0

dα

∫ ∞0

dβsinh2(α) sinh2(β)

(cosh(α) + cosh(β))6.

Problem 107. Calculate the double integral

12π2

∫ π

0

∫ π

0

dαdα′ ln(2− cos(α)− cos(α′))

utilizing the identity

cos(α) + cos(α′) ≡ 2 cos(α+ α′

2

)cos(α− α′

2

)the transformation x = (α+ α′)/2, y = (α− α′)/2 and the integral∫ π

0

ln(1 + sin(x))dx = −π ln(2) +G

where G is the Catalan constant.

Problem 108. Let a > 0 and R ≥ 0. Find

Ia(R) =∫ R

0

∫ R

0

dxdy

(x− y)2 + a2≡∫ R

0

∫ R

0

dxdy

x2 + y2 − 2xy + a2.

Problem 109. Calculate the integral∫ 1

0

∫ 1

0

∫ 1

0

∫ 1

0

∫ 1

0

∫ 1

0

dx1dx2dx3dx4dx5dx6

1 + x1 + x2 + x3 + x4 + x5 + x6.

Problem 110. Let n1, n2 ∈ Z. Consider a function f : R2 → R with aperiod 2π for both x1 and x2

f(x1, x2) =∞∑

n1,n2=−∞cn1,n2e

i(n1x1+n2x2)

Integration 141

and ∫ 2π

0

∫ 2π

0

f(x1, x2)dx1dx2 = 0.

Then

cn1,n2 =1

4π2

∫ 2π

0

∫ 2π

0

f(x1, x2)e−i(n1x1+n2x2)dx1dx2

(i) Letf(x1, x2) = sin(x1) + cos(x2).

Find cn1,n2 .(ii) Let

f(x1, x2) = sin(x1) cos(x2).

Find cn1,n2 .

Problem 111. Find a non-negative analytic function f : R2 → R suchthat ∫

R

∫Rf(x, y)dxdy = 1

and ∫R

∫Rx2dxdy =

12,

∫R

∫Ry2dxdy =

12.

Problem 112. Consider the compact set

S := (x, y) : y ≥ x2, y ≤√x, x, y ≥ 0 .

Thus S ⊂ [0, 1]× [0, 1]. Find

I(S) =∫S

dµ =∫S

dxdy.

Problem 113. Let a > 0. Consider the compact set (lemniscate)

S := (x, y) : (x2 + y2)2 ≤ 2a2xy .

FindI(S) =

∫S

dµ =∫S

dxdy.

Introduce polar coordinates x(r, φ) = r cos(φ), y(r, φ) = r sin(φ). Thus wehave

x2 + y2 = r2, xy = r2 cos(φ) sin(φ) ≡ 12

sin(2φ).


Problem 114. Calculate the integrals∫ 1

0

dx2

(∫ 1−x2

0

dx1

)dx2,

∫ 1

0

dx3

(∫ 1−x3

0

dx2

(∫ 1−x3−x2

0

dx1

)).

Extend to n-dimensions. Give an interpretation of the result.

Problem 115. (i) Is the subset of R2

S2 = (x1, x2) : x1 ≥ 0, x2 ≥ 0, x1 + x2 ≤ 1

convex?(ii) Calculate

A =∫S2

dx1dx2.

(iii) The area of a triangle in the plane R2 with vertices at (x1, y1), (x2, y2),(x3, y3) is given by

A = ±12

det

x1 y1 1x2 y2 1x3 y3 1

where the sign is chosen so that the area is nonnegative. Let (x1, y1) =(0, 0), (x2, y2) = (1, 0), (x3, y3) = (0, 1). Find A. Compare to (ii).(iv) Is the subset of R3

S3 = (x1, x2, x3) : x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

convex?(v) Calculate

V =∫S3

dx1dx2dx3.


I =1

8π2

∫ π

−πdα

∫ π

0

sin(β)dβ∫ π

−πF (α, β, γ)dγ.

Find I for F (α, β, γ) = 1, F (α, γ, β) = α+ β + γ, F (α, β, γ) = αβγ.

Problem 117. Let γ > 0. A random variable X is said to be Lorentzianwith parameters (α, γ) if its probability density is given by

fX(x) =1π

γ

(x− α)2 + γ2, x ∈ R.

Integration 143

Let X, Y be two independent Lorentzian random variables with parameters(α, γ) and (β, δ), respectively. Let ε ≥ 0. Show that(a) εX is distributed Lorentzian (εα, εγ)(b) ε+X is distributed Lorentzian (ε+ α, γ)(c) −X is distributed Lorentzian (−α, γ)(d) X + Y is distributed Lorentzian (α+ β, γ + δ)

(e) X−1 is distributed Lorentzian(

αα2+γ2 ,

γα2+γ2

)Problem 118. A random variable X is said to be Lorentzian if its prob-ability density pX is a the form

pX(x) =1π

γ

(x− α)2 + γ2

where γ > 0. We say that X is (α, γ) to indicate that the random variableis Lorentzian with the probability density given above. Let X, Y be twoindependent random variables with (α, γ) and (β, δ). Let ε be a nonnegativereal number. Show that

εX is (εα, εγ)

ε+X is (ε+ α, γ)

−X is (−α, γ)

X + Y is (α+ β, γ + δ)

X−1 is(

α

α2 + γ ∗ 2,

γ

α2 + γ2

).

Problem 119. Consider the set

M = (x1, x2, x3) : x21 + x2

2 ≤ 1 and x21 + x2

3 ≤ 1

which is subset of R3. Obviously M is compact and measureable

I(M) =∫M

dµ =∫

R3χMdµ =

∫R3χMdx1dx2dx3

where χM is the indicator function. Find I(M).

Problem 120. Given a smooth Hamilton function

H(p,q) =n∑j=1

p2j

2+ U(q)


with n degrees of freedom (p = (p1, . . . , pn), q = (q1, . . . , qn). Let V (E) bethe classical phase space volume at energy E of a smooth Hamilton functionis given by

V (E) =∫

R2nΘ(E −H(p,q))dnpdnq

where Θ is the step function. Assume that U(εq) = εmU(q).(i) Consider the transformation

p = E1/2p′, q = E1/nq′

with the inverse transformation

p′ = E−1/2p, q′ = E−1/nq.

Find dnp′dnq′ and H(p′,q′).(ii) Calculate V (E) with the assumption that E > 0. Find the asymptoticbehaviour.

Problem 121. Let a > 0. Find the area of the surface in R3 given bythe intersection of a hyperbolic paraboloid x3 = x1x2/a and the cylinderx2

1 + x22 = a2. We have the parameter representation

x1(r, φ) = r cos(φ), x2(r, φ) = r sin(φ), x3(r, φ) =x1x2

a=

1ar2 cos(φ) sin(φ).

Note that sin(φ) cos(φ) ≡ 12 sin(2φ).

8.2 Programming Problems

Problem 1. Calculate the integral∫ 1

0

| cos(2πx)|dx

using the random numnber generator described in problem 7, chapter 10,page 250, Problems and Solutions in Scientific Computing. Compare to theexact result by solving the integral.

Integration 145


Problem 1. Let α > 0. Show that∫ ∞−∞

sech(αt)dt =2α.

Problem 2. Consider the function φ : R→ R

φ(x) :=

1 for x ∈ [0, 1]0 otherwise

Findψ(x) := φ(2x)− φ(2x− 1).

Draw the function. Calculate ∫ +∞

−∞ψ(x)dx.


e−x2/2 =

12π

∫ ∞−∞

exp(−1

2y2 + ixy

)dy.

Problem 4. Let ` > 0 and r0 > 0. Find the integral∫ r0

0

r3√1 + r2/`2

dr.

Problem 5. Let 0 ≤ r < 1. Consider the Hilbert space L2[0, 2π] andf(θ) ∈ L2[0, 2π]. Show that

12π

∫ 2π

0

f(θ)dθ +1π

∫ 2π

0

∞∑j=1

rjf(θ) cos(j(φ− θ))dθ

=1

2π

∫ 2π

0

f(θ)1− r2

1− 2r cos(φ− θ) + r2dθ.


e−kx sin(ky) sin(kγ)dk

k=

14

ln(x2 + (y − γ)2

x2 + (y + γ)2

).


Note thatf(x) =

∫ ∞0

e−kxdk

k⇒ df

dx= − 1

x.


x2

(x2 + a2)3=

x(x2 − a2)8a2(x2 + a2)3

+1

8a2arctan(x/a).


3π

∫ π/6

0

ln(2 cos(x))dx = 0.338314... .

Problem 9. The content (n-dimensional volume) bounded by a hyper-sphere of radius r is known to be

Vn =2rnπn/2

nΓ(n/2)

where Γ is the gamma function. Let r = 1. Show that

limn→∞

Vn = 0.

Problem 10. Let k = 0, 1, 2, . . . and

yk :=∫ 1

0

xk

1 + x+ x2dx.

Show thatyk+2 + yk+1 + yk =

1k + 1

.

Note that

y0 =∫ 1

0

11 + x+ x2

dx =2√3

arctan(

2x+ 1√3

)∣∣∣∣x=1

x=0

y1 =∫ 1

0

x

1 + x+ x2dx =

12

ln(1 + x+ x2)∣∣∣∣10

− 12

∫ 1

0

11 + x+ x2

dx =12

ln(1 + x+ x2)∣∣∣∣10

− 12y0.

Problem 11. Let c ∈ C with |c| <∞ and <(c) > 0. Consider the integral

I(c) =∫ ∞

0

exp(−cx− x2)dx.

Integration 147

Show that

I(c) =12

∞∑k=0

(−1)kΓ(1/2 + k/2)

k!ck.

Problem 12. Find α > 0 such that∫ ∞0

2αx2e−2αx3/3dx = 1.

Problem 13. Let x = (x1 x2 · · · xn), y = (y1 y2 · · · yn) and x · y =x1y1 + · · ·+ xnyn. Show that

exp(ay2) =1

πn/2

∫Rn

exp(−x2 + 2a1/2x · y)dx1 . . . dxn.

Problem 14. Let b > a. Show that∫ b

a

√y − ab− y

(1− γ

y

)dy

y − x= π

(1− γ

x

√a

b

).

Problem 15. Let c > 0 and k ∈ R. Show that∫ ∞0

exp(−cs2) cos(2ks)ds =12

√π√c

exp(−k2/c).

This integral plays a role in optics.

Problem 16. Let x > 0. Show that

2∫ ∞

0

e−s2ds =

∫ ∞x2

eu√u

Problem 17. Let x ≥ 0. Consider the function f defined by

f(x) =∫ x

0

ln(1 + y)y

dy.

Show that for small X we can write

f(x) = x− x2

4+x3

9− x4

16+ · · ·


Show that

f(x) =π2

6+

12

ln2(x)− f(

1x

).

Give a value of f(1).

Problem 18. Calculating∫cos3(x) sin3(x)dx

student Alice tells you the result is

14

sin4(x)− 16

sin6(x) + C1

and student Bob tells that

16

cos6(x)− 14

cos4(x) + C2

is the result. C1, C2 are the constants of integration. Discuss.

Problem 19. Let f : Rn+1 → R be an analytic function, gj : R → R(j = 1, . . . , n) are analytic functions and c1, . . . , cn are constants. Consider

I(ε) =∫ g1(ε)

c1

dx1

∫ g2(ε)

c2

dx2 · · ·∫ gn(ε)

cn

f(x1, x2, . . . , xn, ε).

Show that

dI(ε)dε

=∫ g1(ε)

c1

dx1

∫ g2(ε)

c2

dx2 · · ·∫ gn(ε)

cn

∂f(x1, x2, . . . , xn, ε)∂ε

+dg1

dε

∫ g2(ε)

c2

dx2

∫c3

g3(ε) · · ·∫ gn(ε)

cn

f(x1, x2, . . . , xn, ε)

+dg2

dε

∫ g1(ε)

c1

dx1

∫ g3(ε)

c3

dx3 · · ·∫ gn(ε)

cn

f(x1, x2, . . . , xn, ε)

+ · · ·

+dgndε

∫ g1(ε)

c1

dx1

∫ g2(ε)

c2

dx2 · · ·∫ gn−1(ε)

cn−1

dxn−1f(x1, x2, . . . , xn, ε).


tdt

e2πt − 1=

14π2

∫ ∞0

τdτ

eτ − 1=

14π2

π2

6=

124.

Integration 149

Problem 21. Let k ∈ N. Show that∫ 2π

0

cosk(θ)dθ = 2(1 + (−1)k)π(k − 1)!!k!!

.

Problem 22. Let k ∈ N. Show that∫ 2π

0

cosk(θ)dθ = 2(1 + (−1)k)π(k − 1)!!k!!

.

Problem 23. Show that∫ 1

0

√1−√xdx =

815.

Hint. Set x = sin4(y). Then dx = 4 sin3(y) cos(y)dy.

Problem 24. Let a > 0. Show that∫x2dx

(a2 − x2)3/2=

x√a2 − x2

− arcsin(x/a) + C


2∫ ∞

0

sin(2px) sin(qx)x

dx = ln|2p+ q||2p− q|

.

Problem 26. For a λ/2 antenna we obtain the expression

E(r, θ) = −ωI0 sin θ4πε0c2r

∫ λ/4

−λ/4cos(k`) sin(ω(t− c−1(r − ` cos θ)))d`.

Calculate E(r, θ).

Problem 27. Let <(a) > 0 and n = 0, 1, . . .. Show that∫ ∞−∞

xne−ax2+pxdx=

∂n

∂pn

∫ ∞−∞

e−ax2+pxdx

=∂n

∂pn

√π

aep

2/(4a).


Problem 28. Let a 6= 0. Show that∫ ∞−∞

sin(ax)x

dx = πsgn(a).

Problem 29. Let c > 1. Show that∫ c

1

sin(x2)dx =12

∫ c2

1

sin(τ)√τdτ.

Problem 30. Show that∫ 1

0

x(1 + x2)1/2dx =13

(23/2 − 1)

setting u = 1 + x2 and hence du = 2xdx.

Problem 31. Let (Bτ , τ ≥ 0) be the linear Brownian motion startingfrom 0. Show that

Γ+ :=∫ 1

0

ds 1(Bs>0)

follows the arcsine distribution, i.e.

P (Γ+ ∈ dτ) =1π

dτ√τ(1− τ)

.

Problem 32. Let a > 0. Show that∫R

sin2(ax/2)x2

dx =12πa.

Problem 33. Let k = 1, 2, . . .. Study the integrals

Lk(bcc) =1π3

∫ π

0

∫ π

0

∫ π

0

(cos(x1) cos(x2) cos(x3))kdx1dx2dx3

Lk(sc) =1π3

13k

∫ π

0

∫ π

0

∫ π

0

(cos(x1) + cos(x2) + cos(x3))kdx1dx2dx3

Lk(fcc) =1π3

13k

∫ π

0

∫ π

0

∫ π

0

(cos(x1) cos(x2)+cos(x2) cos(x3)+cos(x3) cos(x1))dx1dx2dx3.

Integration 151

They play a role in solid state physics for the body-centred cubic lattice,simple cubic lattice, face-center cubic lattice.

Problem 34. Draw the functions

f1(x) = cos(2πx)f2(x) = cos(2π(cos(2πx)))f3(x) = cos(2π(cos(2π(cos(2πx))))).

Extend it to fn(x). Find the integral∫ 2π

0

fn(x)dx.

Problem 35. Show that (Fresnel’s integral)∫ ∞0

cos(x2)dx =∫ ∞

0

sin(x2)dx =√π

2√

2.


1π

∫ ∞−∞

sech(α− β)eiωβdβ = sech(12πω)eiωα.

Problem 37. Let σ > 0. Show that

14σ2

∫ ∞0

x5/2 exp(− x2

8σ2

)dx =

(3210

)1/4

Γ(3/4)σ3/2.

Problem 38. Let α > 0. Show that∫ ∞0

xe−αx2J0(βx)dx =

12αe−β

2/(4α).

Problem 39. Show that ∫ ∞0

dx

cosh3(x)=π

4.

Chapter 9

Functional Equations

A functional equation is any equation that specifies a function in implicitform. An example is the Gamma function Γ(z)

Γ(z + 1) = zΓ(z)

with Γ(1) = Γ(2) = 1. The trigonometric function sin(x) and cos(x) satisfya system of functional equations. So functional equations are equationswhere the unkonws are functions. In most cases it is assumed that thefunctions are continouos.

9.1 Solved ProblemsProblem 1. To solve a number of nonlinear functional equation it ishelpful to have the solution of the linear functional equation

f(x+ y) = f(x) + f(y) (1)

where we assume that the function f is continouos.

Problem 2. Find the solution of the functional equations

f(x+ y) = f(x)f(y) (1)

where we assume that f is continuous.

152

Functional Equations 153

Problem 3. Find the solution of the functional equation

f(xy) = f(x) + f(y) (1)

where we assume that the function f is continuous.

Problem 4. Find the solution of the functional equation

f(xy) = f(x)f(y). (1)

We assume that the function f is continuous.

Problem 5. Find the solution of the Jensen equation

f

(x+ y

2

)=f(x) + f(y)

2(1)

where we assume that f is continuous.

Problem 6. Show that the functional equation

f(x+ y) =f(x) + f(y)1− f(x)f(y)

admits the solutions f(x) = tan(cx), where c is a constant.


f(x+ y) =f(x) + f(y)

1 + f(x)f(y)C2

(1)

admits the solutionf(x) = C tanh(cx). (2)


f(x+ y) =f(x)f(y)f(x) + f(y)

admits the solutionf(x) =

c

x.


f(x+ y) =f(x) + f(y)− 1

2f(x) + 2f(y)− 2f(x)f(y)− 1



11 + tan(cx)

.


f(x+ y) = f(x)f(y) +√f(x)2

√f(y)2 − 1

admits the solutionf(x) = cosh(cx).


f(x+ y + axy) = f(x)f(y)

admits the solutionf(x) = (1 + ax)c.


f(x+ y)− f(x− y) = 4√f(x)f(y)

admits the solutionf(x) = cx2.

Problem 13. Solve the functional equation

f(x+ y) + f(x− y) = 2f(x)f(y)

assuming that f is a continuous function.

Problem 14. Show that the trigonometric functions f(x) = cos(x) andg(x) = sin(x) satisfy the system of functional equations

g(x+ y) = g(x)f(y) + f(x)g(y)f(x+ y) = f(x)f(y)− g(x)g(y)g(x− y) = g(x)f(y)− g(y)f(x)f(x− y) = f(x)f(y) + g(x)g(y).

Problem 15. Show that the Jacobi elliptic functions satisfy the systemof functional equations

f(x± y) =f(x)g(y)h(y)± f(y)g(x)h(x)

1− k2f(x)2f(y)2

Functional Equations 155

g(x± y) =g(x)g(y)∓ f(x)f(y)h(x)h(y)

1− k2f(x)2f(y)2

h(x± y) =h(x)h(y)∓ k2f(x)f(y)g(x)g(y)

1− k2f(x)2f(y)2.

Problem 16. Let a be a positive integer with a ≥ 2. Let 1 ≤ x ≤ a.Consider the equation

g(x− 1)− 2g(x) + g(x+ 1) = −λg(x).

Show that

gj(x) = sin(jπx/(a+ 1)), λj(x) = 2(1− cos(jπ/(a+ 1)))

satisfy this equation.

Problem 17. Let a, c, ε be positive constants. Solve the functional equa-tion

g((θ + c)(mod 1)) = ag(θ) + ε sin(2πθ).




f(x+ y) =f(x) + f(y) + 2f(x)f(y)

1− f(x)f(y)


cx

1− cx.


f(x) = arctan(x).

Show that

f(x) + f(y) = f

(x+ y

1− xy

).

What happens at xy = 1?

Problem 3. Consider the functional equation

g(x) = αg(g(x/α))

with g(0) = 0 and g′(x = 0) = 1. Show that

g(x) =x

1− cx

is a solution with c an arbitarary constant.

Problem 4. We know that

tan(α+ β) ≡ tan(α) + tan(β)1− tan(α) tan(β)

.

Show thattan(α+ β + γ) =

tan(α) + tan(β) + tan(γ)− tan(α) tan(β) tan(γ)1− tan(α) tan(β)− tan(α) tan(γ)− tan(β) tan(γ)

.

Chapter 10

Inequalities

10.1 Solved ProblemsProblem 1. Let a, b ∈ R. Show that

2ab ≤ a2 + b2.

Problem 2. Let a, b ∈ R. Show that

|a+ b| ≤ |a|+ |b|.

Problem 3. Let a, b ∈ R+. Show that

12

(a+ b) ≥√ab.

Problem 4. Let x ≥ 0 and 0 < p < 1. Show that

1p

(1− xp) ≥ 1− x.

Problem 5. Let x ∈ (0, 1). Show that

x(1− x) < x.

157


Problem 6. Let n ∈ N.(i) Show that n < 2n.(ii) Show that n2 < 4n.(iii) Show that if n ≥ 4 then 2n < n!.

Problem 7. Let x, y be two nonnegative real numbers. Show that

xy ≤(x+ y

2

)2

≡ 14

(x2 + y2 + 2xy).

Problem 8. Let a, b, c, d be nonnegative real numbers. Show that

(abcd)1/4 ≤ 14

(a+ b+ c+ d).

Problem 9. Let xj (j = 1, . . . , n) be nonegative real numbers. Showthat

x1x2 · · ·xn ≤(x1 + x2 + · · ·+ xn

n

)n.

Problem 10. Let a, b, c, d be nonnegative real numbers. Show that

a4 + b4 + c4 + d4 ≥ 4abcd.

Problem 11. Let x ∈ R and n ∈ N. Show that (Bernoulli inequality)

(1 + x)n ≥ 1 + nx.

Problem 12. Let x ≥ 0. Show that

1− e−x ≥ x

1 + x.

Problem 13. Let

en := 1 +11!

+12!

+ · · ·+ 1n!

where n = 1, 2, . . .. Let m > n. Show that

|em − en| = em − en ≤2

(n+ 1)!.

Inequalities 159

Problem 14. Does the inequality

1 + 2 cos(θ)− cos(2θ) ≤ 2

hold for all θ ∈ [0, 2π)?


1 +x

2− x2

8≤√

1 + x ≤ 1 +x

2.


1− 1x≤ ln(x) ≤ x− 1

with equality iff x = 1.

Problem 17. Show that if a twice differentiable function f : R→ R hasa second order derivative which is non-negative (positive) everywhere, thenthe function is convex (strictly convex).

Problem 18. Let a1, a2, . . . , an be positive numbers and b1, b2, . . . , bn benonnegative numbers such that

n∑j=1

bj > 0.

Show that (log-sum inequality)

n∑j=1

(aj log

ajbj

)≥

n∑j=1

aj

log

(∑nj=1 aj

)(∑n

j=1 bj

)with the conventions based on continuity arguments

0 · log 0 = 0, 0 · logp

0=∞, p > 0.

Show that equality holds if and only if aj/bj = constant for all j =1, 2, . . . , n.

Problem 19. Let xj (j = 1, 2, . . . , n) be positive real numbers. Showthat

(x1x2 · · ·xn)1/n ≤∑nk=1 xkn

.


This is the arithmetic-geometric mean inequality.

Problem 20. Let n be a positive integer and a, b ≥ c/2 > 0. Show that

|a−n − b−n| ≤ 4nc−n−1|a− b|.

Problem 21. Let X ⊂ R be an interval. A function ψ : X → R isconvex if for all x1, x2 ∈ X and numbers α1, α2 ≥ 0 with α1 + α2 = 1,

ψ(α1x1 + α2x2) ≤ α1ψ(x1) + α2ψ(x2). (1)

This means that every chord of the graph of ψ lies above the graph. Letψ : X → R be convex, let x1, x2, . . . , xn ∈ X, and let α1, α2, . . . , αn ≥ 0satisfy

∑nj=1 αj = 1. Show that (Jensen’s inequality)

ψ

(n∑i=1

αixi

)≤

n∑i=1

αiψ(xi). (2)

Problem 22. Consider the differentiable function f : [0,∞)→ R

f(x) =x

1 + x.

Let a, b ∈ R+. Show that

f(|a+ b|) ≤ f(|a|+ |b|). (1)

Problem 23. Let a, b ∈ R. Show that

|a+ b|1 + |a+ b|

≤ |a|1 + |a|

+|b|

1 + |b|.

Problem 24. Show that f : R→ R, f(x) = |x| is convex.

Problem 25. Let n ∈ N and n ≥ 2. Show that

en ln(n+ 1) < en+1 ln(n).

Problem 26. Show that the function f : (0,∞)→ R

f(x) = x ln(x)

Inequalities 161

is convex.

Problem 27. Let x, y ∈ R. Show that

x+ y

1 + |x+ y|≤ |x|

1 + |x|+|y|

1 + |y|.

Problem 28. Let A, B be n× n matrices over C. Show that

‖AB‖ ≤ ‖A‖ · ‖B‖.


‖A+B‖ ≤ ‖A‖+ ‖B‖.

Problem 30. Let H be a Hilbert space. Let ψ, φ ∈ H. Assume thatψ 6= 0, φ 6= 0. Show that

|〈φ, ψ〉| ≤√〈φ, φ〉

√〈ψ,ψ〉 (1)

where 〈 , 〉 denotes the scalar product in the Hilbert space.

Problem 31. Let n ≥ 2. Let x1, x2, . . . , xn be given positive real numberwith

x1 < x2 < · · · < xn.

Let λ1, . . . , λn ≥ 0 and∑nj=1 λj = 1. Show that n∑

j=1

λjxj

n∑j=1

λjx−1j

≤ A2G−2

whereA =

12

(x1 + xn), G = (x1xn)1/2.

Problem 32. Let x,y ∈ Rn (column vectors) and ε > 0. Show that

2xTy ≤ εxTx +1εyTy.

Problem 33. Let a, b ∈ R and ε > 0. Show that

2ab ≤ εa2 + ε−1b2.


Problem 34. Show that the analytic function f : R2 → R

f(α, β) = sin(α) sin(β) cos(α− β)

is bounded between −1/8 and 1.

Problem 35. Let n be an integer and n ≥ 2. Show that

en ln(n+ 1) < en+1 ln(n).

Problem 36. Let a, b, c be positive numbers. Assume that

x2

a2+y2

b2+z2

c2= 1.

Show thatx+ y + z ≤

√a2 + b2 + c2.

Use the Lagrange multiplier method.

Problem 37. Let a, b, c, d be positive numbers and s := a + b + c + d.Show that

s

s− a+

s

s− b+

s

s− c+

s

s− d≥ 16

3.

Problem 38. Let x ≥ 0 and 0 < p < 1. Show that

1p

(1− xp) ≥ 1− x.

Problem 39. Let b be a real number such that the elements of the infinitesequence (ak)∞k=1 satisfy

ak+m ≤ ak + am + b

for all k,m = 1, 2, . . .. Show that

a := limk→∞

akk

exists and ak ≥ ka− b for all k.


1√1

+1√2

+ · · ·+ 1√n>√n.

Inequalities 163

Problem 41. Let a, b ≥ 0. Show that

a+ b

1 + a+ b≤ a+ b+ ab

1 + a+ b+ ab≤ a

1 + a+

b

1 + b.


1π2

∫ π/2

0

∫ π/2

0

sin2(x− y)(x− y)2

dxdy <14.

Problem 43. Let x1, x2, x3 ∈ R. Show that

x21 + x2

2 + x23 ≥ x1x2 + x2x3 + x3x1.

Why could it be useful to consider the function f : R3 → R

f(x1, x2, x3) = x21 + x2

2 + x23 − x1x2 − x2x3 − x3x1 ?

Problem 44. Let x1, x2, . . . , xn be positive numbers. Assume that thenumbers satisfy

n∑j=1

xj = 1,n∑j=1

x2j = b2.

Show that

maxxj : 1 ≤ j ≤ n ≤ 1n

(1 +√n− 1

√nb2 − 1).

Problem 45. Consider the function f : R→ R, f(x) = x2. Show that

|f(x2)− f(x1)| ≤ 4|x2 − x1|

for all x1 and x2 in [−2, 2]. Hint. Apply

x21 − x2

2 ≡ (x1 + x2)(x1 − x2).

Problem 46. Let n ∈ N. Show that

n

2< 1 +

12

+13

+ · · ·+ 12n − 1

≤ n.


Problem 47. Let

0 ≤ a1 ≤ a2 ≤ · · · ≤ an, 0 ≤ b1 ≤ b2 ≤ · · · ≤ bn

and r ≥ 1. Show that (Chebyshev inequality) 1n

n∑j=1

arj

1/r 1n

n∑j=1

brj

1/r

≤

1n

n∑j=1

(ajbj)r

1/r

.

Problem 48. Let a, x, y ∈ R. Show that

|√a2 + x2 −

√a2 + y2| ≤ |x− y|.

Problem 49. Let b > a > 0. Show that(1− a

b

)< ln

(b

a

)<b

a− 1.

Problem 50. Show that∣∣∣∣∫ 1

0

cos(nx)x+ 1

∣∣∣∣ ≤ ln(2)

for n ∈ N.


2n <(

2nn

)< 22n.

Problem 52. Consider the two manifolds

x21 + x2

2 = 1, y21 + y2

2 = 1.

Show that|x1y1 + x2y2| ≤ 1.

Hint. Set

x1(t) = cos(t), x2(t) = sin(t), y1(t) = cos(τ), y2(t) = sin(τ).

Inequalities 165

Problem 53. Let n ∈ N and x > 0. Show that

xn+1 +1

xn+1> xn +

1xn.

Problem 54. Find all x ∈ R and x > 0 such that

|√x−√

2| < 1.

Problem 55. Let j, k be positive intgers. Find all pairs (j, k) such thatthe following four inequalities are satisfied

j + k < 10, j + k ≥ 6,j

k> 1,

j

k< 2.

Problem 56. Let n be a positive integer. Show that

1√1

+1√2

+ · · ·+ 1√n≥√n.

Problem 57. Let z = x+ iy with x, y ∈ R. Show that

1√2

(|x|+ |y|) ≤ |z| ≤ |x|+ |y|.


Matrices


|tr(AB∗)|2 ≤ tr(AA∗)tr(BB∗).

Problem 59. Let A and B be two n × n matrices over R. It can beshown that

treA+B ≤ tr(eAeB) ≤ 12

tr(e2A + e2B)

treA+B ≤ tr(eAeB) ≤ (trepA)1/p(treqB)1/q

where p > 1, q > 1 with1p

+1q

= 1.

Is

(trepA)1/p(treqB)1/q ≤ 12

tr(e2A + e2B) ?

Prove or disprove.

Problem 60. Let A, B be hermitian matrices. Then

tr(eA+B) ≤ tr(eAeB).

Assume that

A =(

0 00 1

), B =

(0 11 0

).

Calculate the left and right-hand side of the inequality. Does equality hold?

Problem 61. Let A, B be positive definite matrices. Show that

tr(AB)2p+1≤ tr(A2B2)2p , p a non-negative integer

Problem 62. Let A be an n× n matrix with ‖A‖ < 1.(i) Show that (In +A)−1 exists.(ii) Show that

‖(In +A)−1‖ ≤ 11− ‖A‖

.

Inequalities 167

Problem 63. Let A be an n×n matrix over R. Assume that ajj ≥ 1 forall j and

n∑j 6=k

a2jk < 1.

Show that A is inververtible.

Problem 64. Let A be an n × n matrix over C. Let In be the n × nidentity matrix. Assume that

n∑k=1

|ajk| < 1

for each j. Show that In −A is invertible.

Problem 65. Let A be an n× n positive definite matrix over R. Let Bbe an n× n positive semidefinite matrix over R. Show that

det(A+B) ≥ det(A).

Problem 66. Let A be an n × n matrix over R. Show that there existsnonnull vectors x1, x2 in Rn such that

xT1 Ax1

xT1 x1≤ xTAx

xTx≤ xT2 Ax2

xT2 x2

for every nonnull vector x in Rn.

Problem 67. Let A be an n × n skew-symmetric matrix over R. Showthat

det(In +A) ≥ 1

with equality holding if and only if A = 0.

Problem 68. Let A be an n×n matrix over R. Assume that ajj ≥ 1 forj = 1, 2, . . . , n and

n∑j 6=k

a2jk < 1.

Show that A is invertible.

Problem 69. Let A be an n× n matrix over C. Assume that

|ajj | >n∑k 6=j

|ajk|


for all j = 1, 2, . . . , n. Show that A is invertible.

Problem 70. Let A, B be n× n positive definite matries. Show that

tr(A ln(A))− tr(A ln(B)) ≥ tr(A−B).

Problem 71. Let v be a normalized (column) vector in Cn and let A bean n× n hermitian matrix. Is

v∗eAv ≥ ev∗Av

for all normalized v? Prove or disprove.

Inequalities 169



1√n− 1

> 2√n− 2

√n− 1 >

1√n.

Use this reult to show that

1 +1√2

+1√3

+ · · ·

diverges.

Problem 2. Let b > a. Show that

b− a1 + b2

< arctan(b)− arctan(a) <b− a1 + a2

.

Note that arctan(−x) = − arctan(x).

Problem 3. Let n be is positive integer. Isn+1√

(n+ 1)! > n√n!

for all n?

Problem 4. Let a, b, c ∈ C with |a| = |b| = |c| = 1. Suppose that=(a) ≥ 0, =(b) ≥ 0, =(c) ≤ 0. Show that∣∣∣∣−1 + ab+ bc+ ca

4

∣∣∣∣ ≤ 1√2.

Problem 5. Let n be a positive integer with n ≥ 2. Show that

12

+14

+18

+ · · ·+ 12n−1

< 1.

Problem 6. Let n ≥ 2 and x1, x2, . . . , xn be positive numbers. Assumethat the numbers satisfy

n∑j=1

xj = 1,n∑j=1

x2j = b2.

Show that

maxxj : 1 ≤ j ≤ n ≤ 1n

(1 +√n− 1

√nb2 − 1

).



1 +14

+19

+ · · ·+ 1n2≤ 2− 1

n.

Problem 8. Let x ≥ 0. Show that 1 + x ≤ 3x.

Problem 9. Let A be an n× n matrix. Is

|det(In +A)| ≤ exp(‖A‖)?

Does it depend on the chosen norm?

Problem 10. Let zj (j = 1, . . . , p) be fixed complex numbers. Is

p∏j=1

|z − zj | >p∏j=1

(|zj | − |z|) ?

Problem 11. Let x ∈ (0, 1). Show that

x > ln(1 + x) > x− x2

2.

Problem 12. Let x, y ∈ R. Show that

|x+ y|1 + |x+ y|

≤ |x|1 + |x|

+|y|

1 + |y|.

Problem 13. Let x1 > 0, x2 > 0, α1 > 0, α2 > 0 and α1 +α2 = 1. Showthat

xα11 xα2

2 ≤ α1x1 + α2x2.

Apply ln(x).

Inequalities 171

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Index

z-transform, 110

Airy functions, 49algebraic, 88Alternating function, 47Angular distance, 51Arc length, 42Associated Legendre polynomials, 76

Bargmann space, 85Bell polynomials, 86Bernoulli inequality, 158Bernoulli numbers, 1, 85Bernoulli polynomials, 84Bernstein basis polynomials, 77Bernstein polynomials, 77Bessel function, 114Beta function, 136Bethe ansatz, 3Braid-like relation, 76Brillouin function, 87Buchberger algorithm, 89

Cantor function, 4Cantor series, 15Cantor series approximation, 5Cauchy integral theorem, 110Cauchy principal value, 132Cauchy sequence, 103Cesaro sum, 4Chebyshev inequality, 164Chebyshev polynomials, 73Chordal distance, 105Convex, 160Convex function, 56

Convex set, 56convolution integral, 119Coulomb potential, 47Critcal points, 74Cubic B-spline, 119Cubic equation, 94

Dawson’s integral, 130Diffeomorphism, 28Diffusion equation, 122Dirichlet integral identity, 123Dirichlet kernel, 23Dominant tidal potential, 75

Elementary symmetric functions, 75Euler dilogarithm function, 50Euler summation formula, 1

Fejer kernel, 8Fermat number, 21Fibonacci numbers, 15, 37Fixed points, 109fixed points, 74Fresnel’s integral, 151

Gamma function, 22, 53, 68Gauss sum, 12

Haar wavelet, 122Hankel matrix, 82, 83Harmonic series, 1Hermite polynomial, 74Hermite polynomials, 73Hilbert transform, 132–134Hurwitz zeta-function, 68

177

178 Index

Invariant, 77

Jensen equation, 153

Kernel, 44Kustaanheimo-Stiefel transformation,

39

Laguerre polynomials, 73Legendre map, 36Legendre polynomials, 75Lemniscate, 141log-sum inequality, 159Logarithmic derivative, 42Lorenz model, 79Lotka Volterra model, 80

Newton’s method, 27, 29

Pade approximant, 64Partition, 30Poisson summation formula, 16Poisson wavelet, 122Polar coordinates, 141Polar form, 110Principal argument, 109Principal value, 133, 134Prolate spheroidal coordinates, 47Pyramide, 10

Quartic equation, 87Quintic equation, 84

Radon transform, 133Recursion relation, 75recursion relation, 89Residue theorem, 111Resultant, 82Rudin-Shapiro substitution, 36

Schwarzian derivative, 60Sonine polynomials, 87Spherical cosine theorem, 75Star product, 32Sum rule, 74

Symmetric group, 80

Toroidal coordinates, 47Trace, 78

Vector product, 35

Wronksi determinant, 45

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