Guide to the Problem Sheets for Analysis 2B · 2020-02-26 · Guide to the Problem Sheets for Analysis 2B Each Problem Sheets consists of 4 parts: Problem Class, Tutorial, Homework,

Guide to the Problem Sheets for Analysis 2B

Each Problem Sheets consists of 4 parts: Problem Class, Tutorial, Homework, Additional Problems.

The Additional Problems typically connect the material in this module to other (past or future)modules. Questions on the exam will not assume that you have done any of these additional problems.

Each problem also has a symbol attached to it. Their meaning is as follows:

• , and - means that the problem tests basic understanding of the definitions and theorems. Onthe exam, this would be a Section A or “easy” Section B question. The difference between the twosymbols is as follows:

– , are problems of a type where there is the possibility of straightforward but non-trivialmodification;

– - are not of such a type.

This difference will matter for revision (where , problems are more relevant), but not so muchduring the semester.

• � means that the problem goes beyond testing basic understanding of the definitions and theorems.On the exam, this would be a “harder” Section B question.

• � means that the problem is harder than anything that I would ask on the exam.

• means that the problem connects to other modules.

Problem Class 1 MA20219 Analysis 2B

- 1. Let f : R→ R be given by f(x) = 1 + 2(x− 1) + 5(x− 1)2. Using only the definition of derivativeexpressed in terms of the first order Taylor formula, determine f ′(1).

- 2. Let X and Y be normed spaces, let f : X → Y be a continuous linear operator and let x0 ∈X .Prove that f ′(x0) = f [meaning that f ′(x0)v = f(v) for all v ∈X ].

- 3. Let f : R2 → R2 be given by

f

([x1

x2

])=

[x1 + 3x2

2x1 + 5x2

].

Let x0 ∈ R2. Determine f ′(x0).

, 4. Show that f : R2 → R given by f ([ x1x2 ]) = x1 cos(x2) is differentiable on R2.

� 5. Consider the function f : R2 → R defined by

f

([x1

x2

])=

{x1x22x21+x42

x1 6= 0

0 x1 = 0.

a) Prove that all directional derivatives of f at 0 exist.

b) Prove that f is not differentiable at 0.

Tutorial 1 MA20219 Analysis 2B

, 6. Let f : R2 → R be given by

f

([x1

x2

])= cos(x1) + x1 sin(x2).

Let x0 ∈ R2. Determine the gradient of f at x0.

� 7. Let f : R2 → R be given by

f

([x1

x2

])=

{x1x2x21+x22

[ x1x2 ] 6= [ 00 ]

0 [ x1x2 ] = [ 00 ] .

(a) Show that both partial derivatives of f exist on R2 and compute these explicitly.

(b) Show that f is not continuous at 0.

- 8. Let Y be a normed space, let f : D ⊂ R→ Y and let x0 ∈ R. Show that f is differentiable at x0

if and only if x0 in an interior point of D and

limx→x0

f(x)− f(x0)

x− x0

exists and that moreover, we have that the above limit equals A1 where A is as in the definition ofdifferentiable.

9. Let f : R → R be given by f(x) = 3 sin(x). Use differential calculus for functions of one variable(from e.g. Analysis 1) and Problem 8 to determine f ′(0) (interpreted in the sense of Analysis 2Brather than that of Analysis 1).

Homework 1 MA20219 Analysis 2B


f

([x1

x2

])= x2

1 + 2x1x2.

Let x0 ∈ R2. Determine the gradient of f at x0.

, 11. Show that f : R2\{0} → R given by f ([ x1x2 ]) = x1x21+x22

+ ex1 sin(x2) is differentiable on R2\{0}.


f

([x1

x2

])=

{x1 + x2 x1x2 = 0,

1 x1x2 6= 0.

a) Prove that both partial derivatives of f at 0 exist;

b) Prove that there exists a direction for which the directional derivative of f at 0 does not exist.

Additional Problems 1 MA20219 Analysis 2B

� 13. Let Z be a linear space. Recall that two norms ‖ · ‖1 and ‖ · ‖2 on Z are called equivalent if thereexist m,M > 0 such that for all z ∈ Z

m‖z‖1 ≤ ‖z‖2 ≤M‖z‖1.

Let (X , ‖·‖1) and (X , ‖·‖2) have equivalent norms and let (Y , ‖·‖1) and (Y , ‖·‖2) have equivalentnorms. Let x0 ∈X and f : X → Y . Show that f : (X , ‖ · ‖1)→ (Y , ‖ · ‖1) is differentiable at x0

if and only if f : (X , ‖ · ‖2)→ (Y , ‖ · ‖2) is differentiable at x0.

14. Let f : R2 → R be given by

f

([x1

x2

])= x2

1 + 3x1 + x2.

Show that f is differentiable at 0 and determine the matrix representation of f ′(0) : R2 → R withrespect to the basis {[ 1

1 ] , [ 32 ]} of R2 (and 1 of R).



f

([x1

x2

])=

x1x2(x2

1 − x22)

x21 + x2

2

x 6= 0

0 x = 0.

Show that ∂2f∂x2∂x1

(0) and ∂2f∂x1∂x2

(0) both exist, but are not equal.


f

([x1

x2

])=

{x1 x2 = x2

1,

0 x2 6= x21.

(a) Show that f is continuous at zero.

(b) Show that all directional derivatives of f at zero exist.

(c) Show that f is not differentiable at zero.


- 17. Define h : (0,∞)× (−π, π)→ R2 by

h

([rϕ

])=

[r cosϕr sinϕ

].

Let f : R2 → R2 be differentiable on R2. Prove that g : (0,∞)× (−π, π)→ R2 defined by g := f ◦his differentiable on (0,∞)×(−π, π) and determine the Jacobian matrix of g in terms of the Jacobianmatrix of f .


f

([x1

x2

])= x2 cos(x1) + x3

2.

Show that f is twice differentiable on R2.


f

([x1

x2

])=

{x21x2−x32x21+x22

x 6= 0

0 x = 0.

Compute all the first and second order partial derivatives of f at zero (or show non-existence ofthese).


- 20. Let X be a normed space, let f : D ⊂X → R2 and let x0 ∈X . Define for j ∈ {1, 2} the functionfj : D ⊂X → R by

f(x) =

[f1(x)f2(x)

].

Show that f is differentiable at x0 if and only for all j ∈ {1, 2} the functions fj are differentiableat x0. Moreover, show that the relationship between the derivatives is:

f ′(x0) =

[f ′1(x0)f ′2(x0)

].

, 21. Let f : R2 → R2 be given by

f

([x1

x2

])=

[x2

1x2 + 3x2

cosx1 + ex2

].

(a) Calculate the Jacobian matrix of f at x0 := [ x01x02 ].

(b) Show that f is differentiable on R2.

- 22. Let f : R2 → R2 be given by

f

([x1

x2

])=

[x1 sin(x2)

x1ex2 + cos(x2)

].

Determine the second order Taylor polynomial of f at zero (i.e., determine the function x 7→f(0) + f ′(0)x+ 1

2f′′(0)(x, x)).

- 23. A function f : D ⊂ Rn → R is called harmonic at x0 ∈ D if f is twice differentiable at x0 and∑nk=1

∂2f∂x2k

(x0) = 0. For W ⊂ D, the function f is called harmonic on W if f is harmonic at all

x0 ∈W .

Prove that f : R2\{0} → R defined by

f

([x1

x2

])= log

(√x2

1 + x22

),

is harmonic on R2\{0}.


24. Let g : Rn → R be continuously differentiable on Rn and define f : C([0, 1];Rn)→ R by

f(x) =

∫ 1

0g(x(s)) ds.

Let x0, v ∈ C([0, 1];Rn). Show that

f ′(x0; v) =

∫ 1

0g′(x0(s)) v(s) ds.

[In your solution you might want to interchange a limit and an integral; you may do so withoutjustifying this.]

25. (a) Let g : R2 → R be continuously differentiable on R2 and define f : C1([0, 1]) × C([0, 1]) → Rby

f(x, u) =

∫ 1

0g(x(s), u(s)) ds.

Define the subspace

W := {(x, u) ∈ C1([0, 1])× C([0, 1]) : x′ = u, x(0) = x(1) = 0},

and the affine space

V := {(x, u) ∈ C1([0, 1])× C([0, 1]) : x′ = u, x(0) = a, x(1) = b},

where a, b ∈ R are given. Assume that f restricted to V has a minimum at (x0, u0). Showthat then

∂g

∂x(x0(s), u0(s))− d

ds

∂g

∂u(x0(s), u0(s)) = 0, ∀s ∈ [0, 1].

This equation is called the Euler–Lagrange equation.[Problems like this will appear in MA30061 Optimal Control (where instead of x′ = u morecomplicated differential equations are considered), MA30059 Mathematical Methods 2 andMA40048 Analytical & geometrical theory of differential equations.][You need not justify the calculations which you do in solving this problem.]

(b) The area of the surface of revolution obtained by revolving the function x : [0, 1]→ R aroundthe horizontal axis is given by

2π

∫ 1

0x(s)

√1 + x′(s)2 ds.

We are interested in finding the curve x which minimizes this surface area and satisfies x(0) =x(1) = cosh(1/2). This is a problem as considered in part (a) with a = b = cosh(1/2) andg(x, u) = x

√1 + u2. Write down the Euler–Lagrange equation for this problem and verify that

x(s) = cosh(s− 12) is a solution.

26. A differentiable function f : D ⊂ R2 → R2 is called irrotational if, writing f(x) =[f1(x)f2(x)

], we

have ∂f1∂x2

= ∂f2∂x1

on D. Define g := f ◦ h (with h the “polar coordinates to Cartesian coordinatestransformation” from Problem 17).

(a) Determine necessary and sufficient conditions on the partial derivatives of g for f to be irro-tational.

(b) Show that g : (0,∞)× (−π, π)→ R2 defined by

g

([rϕ

])=

[− sin(ϕ)

rcos(ϕ)r

]

corresponds to an irrotational f .

[Irrotational functions will appear in MA20223 Vector Calculus and Partial Differential Equations.]

27. A differentiable function f : D ⊂ R2 → R2 is called divergence-free if, writing f(x) =[f1(x)f2(x)

], we

have ∂f1∂x1

+ ∂f2∂x2

= 0 on D. Define g := f ◦h (with h the “polar coordinates to Cartesian coordinatestransformation” from Problem 17).

(a) Determine necessary and sufficient conditions on the partial derivatives of g for f to bedivergence-free.

(b) Show that g : (0,∞)× (−π, π)→ R2 defined by

g

([rϕ

])=

[− sin(ϕ)

rcos(ϕ)r

]

corresponds to a divergence-free f .

[Divergence-free functions will appear in MA20223 Vector Calculus and Partial Differential Equa-tions.]

� 28. Define h : (0,∞)× (−π, π)→ R2 by

h

([rϕ

])=

[r cosϕr sinϕ

],

and q : R2\ ((−∞, 0]× {0})→ (0,∞)× (−π, π) by

q

([x1

x2

])=

[√x2

1 + x22

p(x)

],

where p : R2\ ((−∞, 0]× {0})→ (−π, π) is given by

p

([x1

x2

])=

arctan x2

x1x1 > 0

π2 − arctan x1

x2x2 > 0

−π2 − arctan x1

x2x2 < 0.

(a) Show that h is differentiable on (0,∞)× (−π, π) and calculate its Jacobian matrix.

(b) Show that

arctan y + arctan1

y=

{π2 y > 0

−π2 y < 0,

and use this to show that the definition of p is consistent.

(c) Show that q is differentiable on R2\ ((−∞, 0]× {0}) and calculate its Jacobian matrix.

(d) Show that h and q are inverse of each other.

(e) Verify directly that the Jacobian matrix of q at h ([ r0ϕ0 ]) and the Jacobian matrix of h at [ r0ϕ0 ]are inverses of each other. Explain how we could have instead deduced this from the chainrule.


29. Let B : R2 × R2 → R be a (general) symmetric bilinear form given by

B

([x1

x2

],

[y1

y2

])= ax1y1 + bx2y1 + bx1y2 + cx2y2.

(a) Prove that B is positive definite if and only if a > 0 and ac− b2 > 0.

(b) Prove that B is negative definite if and only if a < 0 and ac− b2 > 0.

(c) Prove that if ac− b2 < 0, then B is neither positive semi-definite nor negative semi-definite.

, 30. Determine the local maxima and minima of f : R2 → R given by

f

([x1

x2

])= x2

1 + x22 + x1 + x2 + x1x2.

- 31. Let f : C → C be given by f(z) = z3. Determine the corresponding real function fR : R2 → R2

obtained by canonically identifying C and R2.

, 32. Define the function f : C→ C as the complex function corresponding to the real function

fR

([x1

x2

])=

[sinx1 coshx2

cosx1 sinhx2

].

Prove that f is complex differentiable on C. This function is called the complex sine function andwill be denoted by z 7→ sin z.



f

([x1

x2

])= x3

1 + x21x2 − x2

2 − 4x2.

, 34. Show that f : C → C given by f(z) = z (the complex conjugate) is not complex differentiable atany point in C.

, 35. Show that f : C→ C given by f(z) = |z|2 is complex differentiable in zero, but not holomorphic inzero.



f

([x1

x2

])= 2x3

1 − 150x1 + 6x1x22 − 3x3

2.

- 37. Let f : D ⊂ C→ C and let z0 ∈ D. Show that f is complex differentiable at z0 if and only if z0 isan interior point of D and there exists a a ∈ C and a g : D → C such that for all z ∈ D

f(z) = f(z0) + a(z − z0) + |z − z0| g(z), limz→z0

g(z) = 0,

and that in that case a is unique and equals f ′(z0).

, 38. Define the function f : C→ C as the complex function corresponding to the real function

fR

([x1

x2

])=

[cosx1 coshx2

− sinx1 sinhx2

].

Prove that f is complex differentiable on C. This function is called the complex cosine functionand will be denoted by z 7→ cos z.


� 39. (a) Let X and Y be normed spaces and let B : X ×X → Y be a continuous bilinear function.Define f : X → Y by

f(x) := B(x, x).

Show that f is differentiable on X and that for x0 ∈X

f ′(x0) = v 7→ B(x0, v) +B(v, x0).

(b) Let f : R2 → R be given by

f

([x1

x2

])= x2

1 + 2x1x2.

Let x0 ∈ R2. Determine f ′(x0) without utilizing partial derivatives [Hint: use part (a)].

(c) Let X be an inner-product space. Show that f : X → R defined by f(x) = ‖x‖2 is differen-tiable on X and determine its derivative at x0 ∈X .

40. Show that if X is finite-dimensional, then any symmetric positive definite bilinear form B : X ×X → R is strictly positive definite.[You may use without proof that such a bilinear form is continuous.]

41. Give an example of an infinite-dimensional normed space X and a symmetric bilinear form B :X ×X → R which is positive definite, but not strictly positive definite.[Infinite-dimensional spaces will be considered in MA40256 Analysis in Hilbert spaces and inMA40057 Functional analysis.]

42. Let W ⊂ Rn be open and such that W is compact.

(a) Let A ∈ Rn×n be a symmetric matrix. Show that if the trace of A is positive, then A cannotbe negative semi-definite.

[Hint: recall from linear algebra how both of these notions relate to the eigenvalues of A].

(b) Let g : W ⊂ Rn → R be continuous on W , twice differentiable on W and such that∑nj=1

∂2g∂x2j

(x) > 0 for all x ∈W .

(i) Show that g cannot have a maximum in W .

(ii) Show that maxx∈W g(x) = maxx∈∂W g(x).

(c) Let f : W ⊂ Rn → R be continuous on W , twice differentiable on W and such that∑nj=1

∂2f∂x2j

(x) = 0 for all x ∈W .

(i) Show that maxx∈W f(x) = maxx∈∂W f(x).[Hint: consider the function g(x) := f(x) + εex1 for ε > 0].

(ii) Show that if f = 0 on ∂W , then f = 0 on W .

(d) Show that for given functions φ : W → R and ψ : ∂W → R the boundary value problem

n∑j=1

∂2u

∂x2j

(x) = φ(x), x ∈W,

u|∂W = ψ,

has at most one solution u : W ⊂ Rn → R which is continuous on W and twice differentiableon W .

[This technique for studying partial differential equations is called the maximum principle and isconsidered further in MA30059 Mathematical methods 2 and MA40203 Theory of partial differentialequations.]


, 43. Determine whether or not the following sets are contour-connected (you may use geometric argu-ments)

(a) W = B2(0) ∪B2(1);

(b) W = B1(0) ∪B1(2);

(c) W = B1(0)\{0};(d) W = [0, 1];

(e) W = R\{0};(f) W = {z ∈ C : Re(z) ∈ [−1, 1] or Im(z) ∈ [−1, 1]}; (here as usual “or” is non-exclusive)

- 44. Let γ : [0, 2π]→ C be given by γ(t) = eit. Calculate∫γz dz.

� 45. For R > 0, let γR : [0, π]→ C by defined by γR(t) = Reit.

(a) Let g : Γ→ C be continuous. Prove Jordan’s Lemma:∣∣∣∫γR eizg(z) dz

∣∣∣ ≤ πmaxz∈ΓR|g(z)|.

(b) Show that limR→∞∫γR

eiz

z dz = 0.

(c) Show that a “naive” application of the ML-inequality does not give the above result.



(a) W = B1(0);

(b) W = {z ∈ C : z = reiϕ with r > 0, ϕ ∈ (−π/4, π/4) ∪ (3π/4, 5π/4)};(c) W = {z ∈ C : z = reiϕ with r > 0, ϕ ∈ (−π/4, π/4) ∪ (3π/4, 5π/4)} ∪ {0}.

- 47. Let z0, z1 ∈ C and let γ : [0, 1]→ C be given by γ(t) := (1− t)z0 + tz1. Calculate∫γz dz.

, 48. For R > 0, let γR : [0, π] → C by defined by γR(t) = Reit. Show that limR→∞∫γR

eiz

z2+1dz = 0.

[You may use without proof that for the complex exponential the usual exponential rule holds, i.e.that ew1+w2 = ew1ew2 holds for all w1, w2 ∈ C.]



(a) W = C;

(b) W = R;

(c) W is the closed unit square (i.e. the square with vertices 0, 1, 1 + i and i);

(d) W is the complement of the closed unit square;

(e) W = C\R;

(f) W = C\{0}.

- 50. In the ML-inequality lemma, the length of the contour γ : [a, b]→ C was defined as

L :=

∫ b

a|γ′(t)| dt.

In the following cases calculate L and verify that this indeed coincides with what we geometricallywould consider to be the length of the contour: (here z0, z1 ∈ C, R > 0 and α, β ∈ [0, 2π] withα < β):

(a) γ : [α, β]→ C given by γ(t) = z0 +Reit;

(b) γ : [0, 1]→ C given by γ(t) = z0 + t(z1 − z0).


51. For a differentiable function g : D ⊂ R2 → R2 where we denote g(x) =[g1(x)g2(x)

], the divergence is

defined as ∂g1∂x1

+ ∂g2∂x2

and the curl is defined as ∂g2∂x1− ∂g1

∂x2. The function g is called divergence free

(or incompressible) on a set W ⊂ D if its divergence is zero on W and curl free (or irrotational) ifits curl is zero on W .

Let f : D ⊂ C → C and let u, v : D ⊂ R2 → R be such that f(x1 + ix2) = u(x1, x2) + iv(x1, x2)

for all x1 + ix2 ∈ D. Define g : D ⊂ R2 → R2 by g(x) =[u(x)−v(x)

](then g is the real function

corresponding to the complex conjugate of f). Show that f is complex differentiable on W if andonly if g is real differentiable on W and is both incompressible and irrotational on W .

52. Let g : R2\{0} → R2 be given by

g

([x1

x2

])=

−x2

x21 + x2

2x1

x21 + x2

2

.Use Problem 51 to show that g is divergence free and curl free.

53. In this problem we connect complex and real contour integrals. We first define the notion of a realcontour integral (also known as the “work integral of a vector field”).

Definition. Let γ : [a, b]→ R2 be a contour with image Γ and let P,Q : Γ→ R be continuous. Thereal contour integral ∫

γP (x, y)dx+Q(x, y)dy,

is defined as ∫ b

aP (x(t), y(t))

dx

dt+Q(x(t), y(t))

dy

dtdt,

where γ = (x, y).

(a) Let f : D ⊂ C → C, let u, v : D ⊂ R2 → R be such that f(x + iy) = u(x, y) + iv(x, y) for allx + iy ∈ D and let γ : [a, b] → C be a contour with image Γ. Let γR : [a, b] → R2 be the realfunction corresponding to γ. Assume that f is continuous on Γ. Show that∫

γf(z) dz =

∫γR

u(x, y) dx− v(x, y) dy + i

∫γR

v(x, y) dx+ u(x, y) dy.

The following is a result which will appear in MA20223: Vector Calculus and Partial DifferentialEquations (though only a very special case will be proven there).

Theorem (Green’s Theorem). Let γR be a circuit which encloses the set S (with positive orienta-tion) and let P,Q : D → R be continuously differentiable on γR ∪ S. Then∫

γR

P (x, y)dx+Q(x, y)dy =x

S

∂Q

∂x− ∂P

∂ydxdy.

(b) Let γ be a circuit which encloses the set S (with positive orientation) and let f be such thatfR is continuously differentiable on γR ∪ S. Write down an expression involving real doubleintegrals which equals

∫γ f(z) dz.

(c) Let γ be a circuit which encloses the set S (with positive orientation) and let f be continuouslycomplex differentiable on γ ∪ S. Determine

∫γ f(z) dz.

54. Let S be the unit square {z ∈ C : Re(z), Im(z) ∈ [0, 1]} and let γ be its boundary with positiveorientation (i.e. the vertices are traversed in the order 0, 1, 1 + i, i). Use Problem 53 to determinethe following integrals

(a)∫γ z dz;

(b)∫γ z dz.

55. Let Γ be the boundary of the triangle with vertices 0, 1 and i (traversed in that order). Calculate(using the convention from the lecture notes about how an oriented polygon is seen as a circuit)∫

Γz dz,

(a) using Problem 47;

(b) using Problem 53.

56. Let γ : [a, b] → C be continuously differentiable and let φ : [c, d] → [a, b] be a continuouslydifferentiable bijection with φ′(t) > 0 for all t ∈ [c, d]. Define γ1 : [c, d] → C by γ1 := γ ◦ φ. Provethat

(a) the images of γ1 and γ are equal;

(b) γ1 is continuously differentiable [you may assume the chain rule for complex valued functionswithout proof];

(c) the lengths of the contours are equal (the length of a contour σ : [α, β] → C is defined as∫ βα |σ

′(t)| dt);(d) for all f : Γ→ C continuous (where Γ is the image of both γ and γ1) there holds

∫γ f(z) dz =∫

γ1f(z) dz.


- 57. Let f : C\(−∞, 0]→ C. Define u, v : (0,∞)× (−π, π)→ R through

f(reiϕ) = u

([rϕ

])+ iv

([rϕ

]).

Let z0 = r0eiϕ0 with r0 > 0 and ϕ0 ∈ (−π, π). Show that f is complex differentiable at z0 if andonly if u and v are real differentiable at [ r0ϕ0 ] and

∂u

∂r

([r0

ϕ0

])=

1

r0

∂v

∂ϕ

([r0

ϕ0

]),

∂v

∂r

([r0

ϕ0

])=−1

r0

∂u

∂ϕ

([r0

ϕ0

]).

Further show that in that case

f ′(z0) =

(∂u

∂r

([r0

ϕ0

])cosϕ0 +

∂v

∂r

([r0

ϕ0

])sinϕ0

)+i

(−∂u∂r

([r0

ϕ0

])sinϕ0 +

∂v

∂r

([r0

ϕ0

])cosϕ0

).

[Hint: use Problems 17 and 28].

- 58. Define the function f : C\(−∞, 0] → C through f(reiϕ) = u ([ rϕ ]) + iv ([ rϕ ]) with u, v : (0,∞) ×(−π, π) → R defined by u(r, ϕ) := log r and v(r, ϕ) := ϕ. This function is called (the principalbranch of) the complex logarithm.

(a) Prove that f is complex differentiable on C\(−∞, 0] with derivative z 7→ 1z .

(b) Prove (directly) that f cannot be extended to a function which is continuous on C\{0}.


- 59. Let γ1 : [0, 1]→ C be defined by γ1(t) = −1 + 2t. Let γ2 : [0, π]→ C be defined by γ2(t) = eit.

(a) Show that the concatenation γ1 + γ2 is well-defined.

(b) Calculate∫γ1+γ2

z2 dz.

� 60. From the lecture notes we know that∫γ

1z dz = 2iπ when γ : [0, 2π]→ C is given by γ(t) = eit. Use

this to show that z 7→ 1z does not have an anti-derivative on C\{0}.

� 61. (a) Define a function which is complex differentiable on C\(−∞, 0] and coincides with the usualsquare root function on (0,∞). [Hint: use polar coordinates. Formally applying the exponen-tial rules will tell you what the correct formula should be.]

(b) Define a line segment L from 0 to ∞ which includes the end-point zero but no other realnumbers and a function which is complex differentiable on C\L which coincides with the usualsquare root function on (0,∞) and which has the value i at −1. [Hint: use polar coordinates,but with a different choice for the domain of the angle.]

(c) Define a line segment L from 0 to ∞ which includes the end-point zero but no other realnumbers and a function which is complex differentiable on C\L which coincides with the usualsquare root function on (0,∞) and which has the value −i at −1.

[In Problems 61b and 61c full details of complex differentiability do not have to be given.]


� 62. Define a function which is complex differentiable on C\(−∞, 0] and coincides with the usual cuberoot function on (0,∞). [Hint: use polar coordinates. Formally applying the exponential rules willtell you what the correct formula should be.]

, 63. Let γ : [0, π]→ C be defined by γ(t) = eit. Calculate∫γ ez dz.

, 64. Determine∫∂B3(π) cos z dz.

� 65. Given an example of a function f : W ⊂ C→ C with W open and non-empty which is holomorphicon W with derivative zero, but which is not constant.


� 66. Calculate∫∂B1(0) e1/z dz (here ∂B1(0) denotes the boundary of the unit disc, i.e. the unit circle).

67. Let x0 ∈ R2 and let σ : [−1, 1] → R2 be a contour with σ(0) = x0 and which is differentiableat 0. The tangent vector of σ at x0 is defined as σ′(0). For two contours σ1 and σ2 which aredifferentiable at 0, have nonzero derivative at 0 and have σ1(0) = σ2(0) = x0, we define the anglebetween them at x0 to be the (oriented) angle between their tangent vectors at x0.

A function f : D ⊂ R2 → R2 is called conformal at x0 if the angle between f ◦ γ1 and f ◦ γ2 atf(x0) is the same as the angle between γ1 and γ2 at x0 for all contours γ1, γ2 : [−1, 1] → R2 withγ1(0) = x0 = γ2(0) which are differentiable at 0 with nonzero derivative.

Let f : D ⊂ R2 → R2 and let x0 ∈ D. Assume that f is differentiable at x0 with non-zero Jacobianmatrix. Prove that f is conformal at x0 if and only if the corresponding complex function is complexdifferentiable at z0 (where z0 is the complex number corresponding to x0).

[You should recall from linear algebra which linear functions preserve (oriented) angles.]


� 68. Show that for all ω ∈ R1√2π

∫R

e−x2/2eiωx dx = e−ω

2/2.

[Hint: integrate e−z2/2 around the rectangle with vertices −R, R, R+ iω and −R+ iω and then let

R→∞; you may use without proof that∫R e−x

2/2 dx =√

2π.]


, 69. Determine whether or not the following sets are star-shaped

(i) C;

(ii) C\{0};(iii) C\(−∞, 0].

70. Determine the radius of convergence of the power series∑∞

k=0k6z

k.

71. Determine the radius of convergence of the power series∑∞

k=0(−1)k

2kz2k.

� 72. Find a solution of the differential equation f ′(z) = f(z) by making the assumption that f is given bya power series and obaining a recurrence relation for the coefficients from the differential equation.


, 73. Determine whether or not the following sets are star-shaped

(i) B1(0);

(ii) C\R.

, 74. Calculate∫γ ecos(z) dz where γ is a circuit.

� 75. Show that a star-shaped set is contour-connected.

� 76. Give an example of a contour-connected set which is not star-shaped.

� 77. Use the fact that∫∂Br(z0)

1z−z0 dz = 2iπ from the lecture notes to show that C\{z0} is not star-

shaped.

78. Consider the power series∑∞

k=11k2zk.

(a) Determine the radius of convergence R of the above power series.

(b) Does the power series converge uniformly on BR(0)?

79. Consider the power series∑∞

k=0(2 + (−1)k)kzk.

(a) What happens if we try to apply the ratio test for the radius of convergence?

(b) Determine the radius of convergence.

� 80. Find a solution of the differential equation z2f ′′(z)+zf ′(z)+z2f(z) = 0 by making the assumptionthat f is given by a power series and obaining a recurrence relation for the coefficients from thedifferential equation.

[We note that this equation is the Bessel equation of order zero and appears when solving Laplace’sequation (from MA20223 Vector Calculus and Partial Differential Equations) on a cylinder byseparation of (cylindrical) variables.]


81. (a) By integrating the power series of 11+z with centre z0 = 0 term-by-term, determine the power

series of log(1 + z) with centre z0 = 0.

(b) For θ ∈ (−π, π) show that 1 + eiθ = eiθ/22 cos(θ/2). Use this and Problem 58 to determine theimaginary part of log(1 + eiθ).

(c) By substituting z = eiθ with θ ∈ (−π, π) in the obtained power series, formally obtain theFourier series of the 2π-periodic function defined in (−π, π) by θ 7→ θ

2 , i.e. determine a0 ∈ Rand real-valued sequences (an)∞n=1 and (bn)∞n=1 such that

θ

2∼ a0

2+

∞∑n=1

an cos(nθ) + bn sin(nθ).

[Fourier series are studied in MA20223 Vector Calculus and Partial Differential Equations. Asthe above indicates, Fourier series are closely connected to Power series (and more generally,to Laurent series studied in Chapter 19).]

82. By considering the power series with centre zero of log(i+zi−z

)= log(i+ z)− log(i− z), similarly as

in Problem 81, formally obtain the Fourier series of the 2π-periodic function defined on (−π, π) by{−π/2 θ ∈ (−π/2, π/2)

π/2 θ ∈ (−π,−π/2) ∪ (π/2, π).

� 83. Let f : B1(0)→ C be continuous on B1(0) and holomorphic on B1(0). Show that∫∂B1(0) f(z) dz =

0.

[Hint: first show that z 7→ f(rz) converges uniformly to f on ∂B1(0) as r ↑ 1 by uniform continuityof f .]


, 84. Determine the radius of convergence of the power series of 1z2+1

with centre z0 = 0 without calcu-lating the series.

� 85. Determine∫∂B1(0)

sin zz2

dz.

� 86. Let (fn)∞n=0 with fn : D ⊂ C → C be a sequence of functions, let f : D ⊂ C → C and let W ⊂ Dbe open. Show that if the fn are holomorphic on W and converge uniformly on W to f , then f isholomorphic on W .

� 87. Prove that an entire function f for which there exist n ∈ N0 and C > 0 such that |f(z)| ≤ C|zn|for all z ∈ C is a polynomial of degree at most n.

� 88. Prove that if the entire function f is such that there exists a nonempty open set W ⊂ C such thatf(z) /∈W for all z ∈ C, then f is constant.

� 89. Give an example of a non-constant entire function f and a nonempty set W ⊂ C such that f(z) /∈Wfor all z ∈ C.


, 90. Determine the radius of convergence of the power series of zz2+4

centred at z0 = 0 without calculatingthe series.

� 91. Let W ⊂ C be open, let p ∈W and let f : W ⊂ C→ C be holomorphic on W\{p} and continuouson W . Show that f is holomorphic on W .

� 92. Determine all entire functions f with the property that the imaginary part of f(z) is nonnegativefor all z ∈ C.

� 93. Determine all entire functions f with the property that |f ′(z)| < |f(z)| for all z ∈ C.


, 94. Determine the radius of convergence of the power series of 1z2−1

centred at z0 = 3 without calculatingthe series.

95. (a) Prove that for a non-constant polynomial p we have

lim|z|→∞

|p(z)| =∞.

(b) Prove the Fundamental Theorem of Algebra.

� 96. Let f be an entire function with the property that there exists a M > 0 such that |f(z)| ≤MeRe(z)

for all z ∈ C. Prove that f(z) = Cez for some constant C ∈ C.


97. Let g : [0,∞)→ R be an integrable function and define W := {z ∈ C : Re(z) > 0} and f : W → Cby

f(z) =

∫ ∞0

e−ztg(t) dt.

Prove that f is holomorphic on W .

[You may use Fubini’s Theorem without verifying its assumptions. You may assume without proofthat f is continuous (this is most easily shown using the Dominated Convergence Theorem fromMA40042 Measure theory & integration). The function f is the Laplace transform of g and wasstudied in MA20220 Ordinary Differential Equations and Control.]


� 98. Let f : D ⊂ C → C, let z0 ∈ D and assume that f is holomorphic at z0. For N ∈ N0, show thatf has a zero of order N at z0 if and only if f (n)(z0) = 0 for all n ∈ N0 with 0 ≤ n ≤ N − 1 whilef (N)(z0) 6= 0.

� 99. Use the identity theorem to show that for all z, w ∈ C

ez+w = ezew.

[You may use the fact that the equality is true for z, w ∈ R and that the complex exponentialfunction is holomorphic on C.]


, 100. Determine the order of z0 = 0 as a zero of f : C→ C in case:

(a) f(z) = z;

(b) f(z) = z2;

(c) f(z) = z sin(z);

(d) f(z) = cos(z)

� 101. Show that the set of zeros of the function f : C→ C defined by f(z) = sin 11−z for z 6= 1 and f(1) = 0

has 1 as an accumulation point. Why does this not contradict the isolated zeros theorem?

, 102. Show that there exists exactly one function which is holomorphic on C and coincides with the sinefunction on R.

, 103. Determine whether or not γ := ∂B1(0) is homologous to zero in W when

(a) W := C\{2};(b) W := C\{1/2};(c) W := B5(0)\{3, 1/3}.

, 104. Determine the winding number function of the cycle γ := {∂B1(0), ∂B1(3)}.


, 105. Determine the order of z0 as a zero of f : C→ C in case:

(a) z0 = 1, f(z) = (z − 1)2;

(b) z0 = 0, f(z) = sin2(z).

106. Show the following.

(a) Let S ⊂ C be a finite set. Then all points of S are isolated.

(b) Let S := {nπ : n ∈ Z}. Then all points of S are isolated.

(c) Let S := { 1n : n ∈ N}. Then 0 is an accumulation point of S.

(d) Let S := { 1n : n ∈ N}. Then S does not contain an accumulation point.

(e) Let S := { 1n : n ∈ N} ∪ {0}. Then S contains an accumulation point.

(f) Let S := R. Then S contains an accumulation point.

(g) Let S ⊂ C be a non-empty open set. Then S contains an accumulation point.

, 107. Use the identity theorem to show that there is precisely one function which is holomorphic on Cand coincides with z2 on (0, 1).

� 108. Use the identity theorem to show that for all z, w ∈ C

cos(z + w) = cos z cosw − sin z sinw.

[You may use the fact that the equality is true for z, w ∈ R and that the sine and the cosine areholomorphic on C.]

� 109. Provide a counter-example to the statement of the Minimum Modulus Theorem with the condition0 < |f(z0)| removed.

� 110. Use the Minimum Modulus Theorem to prove the Fundamental Theorem of Algebra.

, 111. Let γ1 = ∂B1(0) and γ2 = ∂B3(0). Determine whether γ1 is homologous to γ2 in W when

(a) W := C\{0};(b) W := C\{2};(c) W := C\{1/2, 2};(d) W := B5(1);

(e) W := Bπ(0)\B1/2(0).

, 112. Let z0, z1, z2 ∈ C and let r > 0 be such that |z1−z0|, |z2−z0| 6= r. Let W := C\{z1, z2}. Determinewhen ∂Br(z0) is homologous to zero in W .


� 113. Prove the following theorem.

Theorem (Maximum modulus theorem for a compact set). Let f : D ⊂ C → C and let W ⊂ Dbe open and contour-connected and such that its closure W is compact. If f is homolorphic on Wand continuous on W , then there exists a z0 ∈ ∂W such that |f(z)| ≤ |f(z0)| for all z ∈W .

- 114. Determine the maxima of the modulus of z2 − 3z + 2 on B1(0).

- 115. Determine the maxima of the modulus of z2 − 1 on B1(0).

� 116. In this question we will use the maximum modulus theorem for a compact set (Problem 113) togive an alternative proof of Liouville’s Theorem. Let f be a bounded entire function and let M besuch that |f(z)| ≤M for all z ∈ C. Define

g(z) =

{f(z)−f(0)

z z 6= 0

f ′(0) z = 0.

(a) Show that g is entire.

(b) Let R > 0. Show that for z ∈ ∂BR(0) there holds

|g(z)| ≤ M + |f(0)|R

.

(c) Use the maximum modulus theorem and a limiting argument to show that g is the zerofunction.

(d) Conclude that f must be constant.

� 117. Define H := {z ∈ C : Re(z) > 0} and H = {z ∈ C : Re(z) ≥ 0}. Let f : H ⊂ C → C beholomorphic on H and assume that f(H) ⊂ H and that f(1) = 0. Show that f ≡ 0.

� 118. Let γ be the square with vertices 0, 2, 2 + 2i, 2i (in that order). Determine the winding numberfunction of γ.

� 119. Let f be a continuous function on the unit circle. Show that f is the uniform limit of a sequence ofpolynomials if and only if f has a continuous extension to the closed unit disc which is holomorphicon the open unit disc.

[Hint: Use Problems 86, 113 and an idea from the proof of Problem Problem 83.]


, 120. In this problem we will calculate the real integral∫∞−∞

cosxx2+1

dx using complex analysis.

For R > 0, let γ1,R = [−R,R], let γ2,R : [0, π] → C by defined by γ2,R(t) = Reit and let γR :=γ1,R + γ2,R.

(a) For R > 1, calculate ∫γR

eiz

z2 + 1dz.

(b) Use the above and Problem 48 to determine∫∞−∞

cosxx2+1

dx.

� 121. Let W ⊂ C be open, let z0 ∈ W and let g, h : D ⊂ C → C be holomorphic on W . Assume that hhas a zero of order 2 at z0. Show that

Res[gh, z0

]=

6h′′(z0)g′(z0)− 2g(z0)h′′′(z0)

3h′′(z0)2.

� 122. In this problem we will calculate the real integral∫∞−∞

1(x2+1)2

dx using complex analysis.

For R > 0, let γ1,R = [−R,R], let γ2,R : [0, π] → C by defined by γ2,R(t) = Reit and let γR :=γ1,R + γ2,R.

(a) For R > 1, calculate ∫γR

1

(z2 + 1)2dz.

(b) Use the above to determine∫∞−∞

1(x2+1)2

dx.


, 123. Let γ := {∂B2(0),−∂B1(0)}. Determine∫γ

sinww2 dw.

, 124. Let γ := {∂B1(0), ∂B1(3)} and let z /∈ Γ. Determine∫γ

cosww−z dw.

, 125. Calculate∫∂B4(0)

ez

sin z dz.

[You may use without proof that all the zeros of the complex sine function are real.]

� 126. Prove the following:

Theorem (Cauchy’s Integral Formula for the derivatives). Let W ⊂ C be open and let f : D ⊂C → C be holomorphic on W . If γ is a cycle in W with image Γ which is homologous to zero inW , then for all z ∈W\Γ and k ∈ N0

nγ(z)f (k)(z) =k!

2iπ

∫γ

f(w)

(w − z)k+1dw.


, 127. Calculate Res[

cos zsin z , 0

].

� 128. In this problem we will calculate the real integral∫∞−∞

cosxcoshx dx using complex analysis.

[You may use without proof that for the complex exponential the usual exponential rule holds, i.e.that ew1+w2 = ew1ew2 holds for all w1, w2 ∈ C. You may use “geometric” arguments to calculatewinding numbers.]

(a) Calculate the zeros of cosh z := ez+e−z

2 .

(b) Calculate Res[

eiz

cosh z , iπ2

].

(c) Calculate ∫ΓR

eiz

cosh zdz,

where ΓR is the boundary of the rectangle with vertices −R, R, R+ iπ and −R+ iπ (in thatorder). Denote the top, bottom, left and right of this rectangle by ΓTR, ΓBR, ΓLR, ΓRR.

(d) Show that the function f(z) := eiz

cosh z satisfies

f(z + iπ) = −e−πf(z),

(e) Use 128d to write the integral of f over ΓTR as a constant times the integral over ΓBR.

(f) Show that |eiz| ≤ 1 for z ∈ {x+ iy : x ∈ R, y ≥ 0}.(g) Show that | cosh z| ≥ | sinhx|, where z = x+ iy and x, y ∈ R.

(h) Conclude that we have |f(z)| ≤ 1| sinhx| for z ∈ {x+ iy : x ∈ R, y ≥ 0, x 6= 0}.

(i) Use the ML-inequality and the result from (128h) to show that

limR→∞

∫ΓLR

f(z) dz = 0 = limR→∞

∫ΓRR

f(z) dz.

(j) Use that cosx = Re eix, that∫ R

−R

eix

coshxdx+

∫ΓRR

f(z) dz +

∫ΓTR

f(z) dz +

∫ΓLR

f(z) dz =

∫ΓR

f(z) dz,

and the above established results to determine∫∞−∞

cosxcoshx dx.

� 129.

Definition. A set W ⊂ C is called simply-connected if it is open, contour-connected and everycircuit in W is homologous to zero in W .

(a) Show that an open contour-connected set W is simply connected if and only if: for all functionsf : W → C which are holomorphic on W and all circuits γ in W there holds

∫γ f(w) dw = 0.

(b) Show that every star-shaped open set is simply-connected.

(c) Given an example of a simply-connected open set which is not star-shaped (you may use ageometric argument).


130. Recall the definition of simply-connected from Problem 129 and the following definition from Prob-lem 23.

Definition. A function u : D ⊂ R2 → R is called harmonic at x0 ∈ D if u is twice differentiableat x0 and ∂2u

∂x21(x0) + ∂2u

∂x22(x0) = 0. For W ⊂ D, the function u is called harmonic on W if u is

harmonic at all x0 ∈W .

Prove that if W is simply-connected with corresponding real set WR, then u : WR ⊂ R2 → R isharmonic on WR if and only if there exists a function f : W ⊂ C→ C which is holomorphic on Wand such that the real function corresponding to f is fR = [ uv ] for some v : WR ⊂ R2 → R.

131. Recall the definition of a harmonic function from Problem 130.

(a) Let V ⊂ C be open and let {zj}mj=1 ⊂ V . Let f : V \{zj}mj=1 ⊂ C → C be holomorphic onV \{zj}mj=1. Let {cj}mj=1 ⊂ R. Let fR = [ u0v0 ] be the real function corresponding to f and let

xj ∈ R2 be the real points corresponding to zj and VR ⊂ R2 the real set corresponding to V .Define u : VR\{xj}mj=1 → R by

u(x) = u0(x) +m∑j=1

cj log ‖x− xj‖. (†)

Show that u is harmonic on VR\{xj}mj=1.

(b) Let VR ⊂ R2 be open and let {xj}mj=1 ⊂ VR. Let u : VR\{xj}mj=1 → R be harmonic onVR\{xj}mj=1. Let {zj}mj=1 ⊂ V be the complex points corresponding to {xj}mj=1. Show thatthere exists a function f : V \{zj}mj=1 ⊂ C → C which is holomorphic on V \{zj}mj=1 andnumbers {cj}mj=1 ⊂ R such that u is given by (†).


� 132. (a) Determine

Res

[π

z2 tan(πz), 0

].

[Hint: use Cauchy’s Integral Formula for the derivatives for the centre of a disc.]

(b) For R ∈ N let γR be the boundary of the square [−R− 12 , R+ 1

2 ]× i[−R− 12 , R+ 1

2 ]. DeterminelimR→∞

∫γR

πz2 tan(πz)

dz.

[You may use without proof that tan(x+ iy) = sin(2x)+i sinh(2y)cos(2x)+cosh(2y) ]

(c) Use (b) to determine∑∞

k=11k2

.

[Hint: use the Residue Theorem. You may use without proof that all the zeros of the complextan function are real and you may use geometric arguments to calculate winding numbers].

� 133. Assume that f has a pole of order one at z0. Let α ∈ [0, 2π]. For ε > 0, define γε : [0, α] → C byγε(z) = z0 + εeit. Show that

limε↓0

∫γε

f(z) dz = αiRes[f, z0].

� 134. Calculate∫R

sinxx dx.

[Hint: write sinxx = Im eix

x , integrate eiz

z over an appropriate circuit and use the application ofJordan’s Lemma from Problem 45 and the result from Problem 133.]


, 135. Consider the function z 7→ 1z(z+1) .

(a) Determine the punctured disc of convergence of the Laurent series with with centre z0 = 0without calculating this Laurent series.

(b) Calculate this Laurent series.

, 136. Classify zero as a removable singularity, pole or essential singularity of the following functions

(a) 1z ,

(b) sin 1z .

� 137. Let p : C→ C be an entire function such that lim|z|→∞ |p(z)| =∞. Prove that p is a polynomial.


, 138. Calculate the Laurent series of the following functions with the given centre and determine itspunctured disc of convergence.

(a) z(z+1)(z+2) with centre z0 = −1,

(b) 2(z−1)(z+1) with centre z0 = 1,

(c) 1z2(z+1)

with centre z0 = −1.

, 139. Classify zero as a removable singularity, pole or essential singularity of the following functions

(a) 1z3− cos z,

(b) ze1/z,

(c) sin zz ,

(d) cos zz .

� 140. Find a two-parameter family of solutions of the differential equation zf ′′(z)+(2−z)f ′(z)−f(z) = 0by making the assumption that f is given by a Laurent series and obaining a recurrence relationfor the coefficients from the differential equation.

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Guide to the Problem Sheets for Analysis 2B · 2020-02-26 · Guide to the Problem Sheets for Analysis 2B Each Problem Sheets consists of 4 parts: Problem Class, Tutorial, Homework,