MATH306 SUPPLEMENTARY MATERIAL A BRIEF …machiang/150/Intro_bessel_bk_Nov08.pdfMATH306 SUPPLEMENTARY MATERIAL A BRIEF INTRODUCTION TO BESSEL and ... 2.3.3 Generating Function for

MATH306 SUPPLEMENTARY MATERIAL

A BRIEF INTRODUCTION TO BESSEL andRELATED SPECIAL FUNCTIONS

Edmund Y. M. Chiangc© Draft date December 1, 2008

Contents

Contents i

1 Trigonometric and Gamma Functions 3

1.1 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Bessel Functions 9

2.1 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Basic Properties of Bessel Functions . . . . . . . . . . . . . . . . . . . 12

2.3.1 Zeros of Bessel Functions . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Recurrence Formulae for Jν . . . . . . . . . . . . . . . . . . . 12

2.3.3 Generating Function for Jn . . . . . . . . . . . . . . . . . . . . 13

2.3.4 Lommel’s Polynomials . . . . . . . . . . . . . . . . . . . . . . 14

2.3.5 Bessel Functions of half-integer Orders . . . . . . . . . . . . . 14

2.3.6 Formulae for Lommel’s polynomials . . . . . . . . . . . . . . . 14

2.3.7 “Pythagoras’ Theorem” for Bessel Function . . . . . . . . . . 15

2.3.8 Orthogonality of the Lommel Polynomials . . . . . . . . . . . 15

2.4 Integral formaulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Asymptotic Behaviours of Bessel Functions . . . . . . . . . . . . . . . 17

2.5.1 Addition formulae . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Fourier-Bessel Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

i

CONTENTS 1

3 An New Perspective 21

3.1 Hypergeometric Equations . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Confluent Hypergeometric Equations . . . . . . . . . . . . . . . . . . 23

3.3 A Definition of Bessel Functions . . . . . . . . . . . . . . . . . . . . . 23

4 Physical Applications of Bessel Functions 27

5 Separation of Variables of Helmholtz Equations 31

5.1 What is not known? . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2 CONTENTS

Chapter 1

Trigonometric and GammaFunctions

1.1 Trigonometric Functions

Pythagoras Theorem:

(1.1) sin2 x+ cos2 x = 1

holds for all real x.

Addition Theorem:

(1.2) sin(A+B) = sinA cosB + sinB cosA

(1.3) cos(A+B) = cosA cosB − sinA sinB

hold for all angles A, B.

1.2 Gamma Function

We recall that

(1.1) k! = k × (k − 1)× · · · 3× 2× 1.

Euler was able to give a correct definition to k! when k is not a positive integer. Heinvented the Euler-Gamma function in the year 1729. That is,

(1.2) Γ(x) =

∫ ∞0

tx−1e−t dt.

3

4 CHAPTER 1. TRIGONOMETRIC AND GAMMA FUNCTIONS

So the integral will “converge” for all positive real x. Since

(1.3) Γ(1) =

∫ ∞0

e−t dt = 1,

so one can show

(1.4) Γ(x+ 1) = xΓ(x),

and so for each positive integer n

(1.5) Γ(n+ 1) = n!.

In fact, the infinite integral will not only “converge” for all positive real x but canbe “analytically continued” into the whole complex plane C. We have

(1.6) Γ(z) =

∫ ∞0

tz−1e−t dt, <(z) > 0,

which is analytic in C except at the simple poles at the negative integers including0. So it is a meromorphic function.

Euler worked out that

(1.7)(1

2

)! =

1

2

√π.

So

(1.8)(

51

2

)! =

11

2

9

2,

7

2,

5

2,

3

2

(1

2

)!.

One can even compute negative factorial:

(1.9) Γ(1/2) = (−1/2)(−3/2)(−5/2)(−19/2)(−11/2)Γ(−11/2).

1.3 Chebyshev Polynomials

Theorem 1.3.1 (De Moivre’s Theorem). Let z = r(cos θ + i sin θ) be a complexnumber. Then for every non-negative integer n,

(1.1) zn = rn(cosnθ + i sinnθ).

Proof It is clear that that the formula holds when n = 1. We assume it holds at theinteger n. Then the addition formulae for the trigonometric theorem imply that

zn+1 = rn+1(cos θ + i sin θ)(cosnθ + i sinnθ)

= rn+1[cos(n+ 1)θ + i sin(n+ 1)θ].

Thus, the result follows by applying the principle of induction. �

1.3. CHEBYSHEV POLYNOMIALS 5

Exercise 1.3.2. Justify the formula for negative integers n.

Remark 1.3.1. De Moivre’s formula actually works for all real and complex n. SeeMATH304.

Example 1.3.3 (Multiple angle formulae). Let n = 2 in De Movire’s formula yields

(1.2) z2 = cos 2θ + i sin 2θ.

But

(1.3) z2 = (cos θ + i sin θ)2 = cos2 θ − sin2 θ + i(2 cos θ sin θ).

Comparing the real and imaginary parts of (1.2) and (1.3) yields

cos 2θ = cos2 θ − sin2 θ

= 2 cos2 θ − 1

= 1− 2 sin2 θ

and

(1.4) sin 2θ = 2 cos θ sin θ.

Similarly, we have

(1.5) z3 = cos 3θ + i sin 3θ.

But

z3 = (cos θ + i sin θ)3

= cos3 θ + 3 cos2 θ(i sin θ) + 3 cos θ(i sin θ)2 + (i sin θ)3

= cos3 θ − 3 cos θ sin2 θ + i(3 cos2 θ sin θ − sin3 θ).(1.6)

Comparing the real and imaginary parts of (1.5) and (1.6) yields

cos 3θ = cos3 θ − 3 cos θ sin2 θ

= 4 cos3 θ − 3 cos θ

and

sin 3θ = 3 cos2 θ sin θ − sin3 θ

= 3 sin θ − 4 sin3 θ.


More generally, we have

cosnθ + i sinnθ = zn = (cos θ + i sin θ)n

=n∑k=0

(n

k

)cosn−k θ(i sin θ)k

=n∑k=0

(n

k

)ik cosn−k θ sink θ.(1.7)

Equating the real parts of the last equation and noting that i2l = (−1)l give

cosnθ = cosn θ −(n

2

)cosn−2 θ sin2 θ +

(n

4

)cosn−4 θ sin4 θ + · · ·

+ (−1)[n/2]

(n

2[n/2]

)cosn−2[n/2] θ sin2[n/2] θ.(1.8)

But each of the sine function above has even power. Thus the substitution ofsin2 θ = 1− cos2 θ in (1.8) yields

(1.9) cosnθ =

[n/2]∑l=0

(n

2l

)cosn−2l θ

( l∑k=0

(−1)k(l

k

)cos2k θ

).

Finally, it is possible to re-write the summation formula as

(1.10) cosnθ =

[n/2]∑k=0

((−1)k

[n/2]∑j=k

(n

2j

)(j

k

))cosn−2k θ.

Writing θ = arccos x for 0 ≤ θ ≤ π, and Tn(x) = cos(n arccosx). We deduce fromthe above formula that Tn(x) is a polynomial in x of degree n. The polynomial Tnis called the Chebyshev1 polynomial of the first kind of degree n. The firstfew polynomials are given by

T0(x) = 1;

T1(x) = x;

T2(x) = 2x2 − 1;

T3(x) = 4x3 − 3x;

T4(x) = 8x4 − 8x2 + 1;

T5(x) = 16x5 − 20x3 + 5x;

· · · · · ·1P. L. Chebyshev (1821–1894) Russian mathematician. The collection of the polynomials

{Tn(x)}∞0 is a complete orthogonal polynomial set on [−1, 1]

1.3. CHEBYSHEV POLYNOMIALS 7

The set of Chebyshev polynomials {Tn(x)} can generate a L2[−1, 1] space. That is,it is a Hilbert space.

Interested reader should consult the handbook [1] and the book [5] about theChebyshev polynomials by T. J. Rivlin. Our library has a copy of this book.


Chapter 2

Bessel Functions

2.1 Power Series

We define the Bessel function of first kind of order ν to be the complex functionrepresented by the power series

(2.1) Jν(z) =+∞∑k=0

(−1)k(12z)ν+2k

Γ(ν + k + 1) k!= zν

+∞∑k=0

(−1)k(12)ν+2k

Γ(ν + k + 1) k!z2k.

Here ν is an arbitrary complex constant and the notation Γ(ν) is the Euler Gammafunction defined by

(2.2) Γ(z) =

∫ ∞0

tz−1e−t dt, <(z) > 0.

2.2 Bessel Equation

A common introduction to the series representation for Bessel functions of first kindof order ν is to consider Frobenius method: Suppose

y(z) =∞∑k=0

ckzα+k,

where α is a fixed parameter and ck are coefficients to be determined. Substitutingthe above series into the Bessel equation

(2.1) z2d2y

dz2+ x

dy

dz+ (z2 − ν2)y = 0.

9

10 CHAPTER 2. BESSEL FUNCTIONS

and separating the coefficients yields:

0 = z2d2y

dz2+ x

dy

dz+ (z2 − ν2)y

=∞∑k=0

ck(α + k)(α + k − 1)zα+k +∞∑k=0

ck(α + k)zα+k

+ (z2 − ν2) ·∞∑k=0

ckzα+k

=∞∑k=0

ck[(α + k)− ν2]zα+k +∞∑k=0

ckzα+k+2

= c0(α2 − ν2)zα +

∞∑k=0

{ck[(α + k)2 − ν2] + ck−2}zα+k

The first term on the right side of the above expression is

c0(α2 − ν2)zα

and the remaining are

c1[(α + 1)2 − ν2] + 0 = 0

c2[(α + 2)2 − ν2] + c0 = 0

c3[(α + 3)2 − ν2] + c1 = 0

· · · · · ·ck[(α + k)2 − ν2] + ck−2 = 0(2.2)

· · · · · ·(2.3)

Hence a series solution could exist only if α = ±ν. When k > 1, then werequire

ck[(α + k)2 − ν2] + ck−2 = 0,

and this determines ck in terms of ck−2 unless α − ν = −2ν or α + ν = 2ν is aninteger. Suppose we discard these exceptional cases, then it follows from (2.2) that

c1 = c3 = c5 = · · · = c2k+1 = · · · = 0.

Thus we could express the coefficients c2k in terms of

c2k =(−1)kc0

(α− ν + 2)(α− ν + 4) · · · (α− ν + 2k) · (α + ν + 2)(α + ν + 4) · · · (α + ν + 2k).

2.2. BESSEL EQUATION 11

If we now choose α = ν, then we obtain

(2.4) c0zν[1 +

∞∑k=0

(−1)k(12z)2k

k!(ν + 1)(ν + 2) · · · (ν + k)

].

Alternatively, if we choose α = −ν, then we obtain

(2.5) c′0z−ν[1 +

∞∑k=0

(−1)k(12z)2k

k!(−ν + 1)(−ν + 2) · · · (−ν + k)

].

Since c0 and c′0 are arbitrary, so we choose them to be

(2.6) c0 =1

2νΓ(ν + 1), c′0 =

1

2−νΓ(−ν + 1)

so that the two series (2.4) and (2.5) can be written in the forms:

(2.7) Jν(z) =+∞∑k=0

(−1)k(12z)ν+2k

Γ(ν + k + 1) k!, J−ν(z) =

+∞∑k=0

(−1)k(12z)−ν+2k

Γ(−ν + k + 1) k!

and both are called the Bessel function of order ν if the first kind.

It can be shown that the Wronskian of Jν and J−ν is given by (G. N. Watson“A Treatise On The Theory Of Bessel Functions”, pp. 42–43):

(2.8) W (Jν , J−ν) = −2 sin νπ

πz.

This shows that the Jν and J−ν forms a fundamental set of solutions when ν isnot equal to an integer. In fact, when ν = n is an integer, then we can easily checkthat

(2.9) J−n(z) = (−1)nJn(z).

Thus it requires an extra effort to find another linearly independent solution.It turns out a second linearly independent solution is given by

(2.10) Yν(z) =Jν(z) cos νπ − J−ν(z)

sin νπ

when ν is not an integer. The case when ν is an integer n is defined by

(2.11) Yn(z) = limν→n

Jν(z) cos νπ − J−ν(z)

sin νπ.


The Yν so defined is linearly independent with Jν for all values of ν.

In particular, we obtain

Yn(z) =−1

π

n−1∑k=0

(n− k − 1)!

k!

(z2

)2k−n(2.12)

+1

π

∞∑k=0

(−1)k(z/2)n+2k

k!(n+ k)!

[2 log

z

2− ψ(k + 1)− ψ(k + n+ 1)

](2.13)

for | arg z| < π and n = 0, 1, 2, · · · with the understanding that we set the sumto be 0 when n = 0. Here the ψ(z) = Γ′(z)/Γ(z). We note that the function isunbounded when z = 0.

2.3 Basic Properties of Bessel Functions

The general reference for Bessel functions is G. N. Watson’s classic: “A Treatise onthe Theory of Bessel Functions”, published by Cambridge University Press in 1922[6].

2.3.1 Zeros of Bessel Functions

See A. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions withformulas, graphs and mathematical tables”, 10th Edt., National Bureau of Stan-dards, 1964. [1]. [3] and [6]

2.3.2 Recurrence Formulae for Jν

We consider arbitrary complex ν.

d

dzzνJν(z) =

d

dz

(−1)kz2ν+2k

2ν+2k k!Γ(ν + k + 1)

=d

dz

(−1)kz2ν−1+2k

2ν−1+2k k!Γ(ν + k)

= zνJν−1(z).

But the left side can be expanded and this yields

(2.1) zJ ′ν(z) + νJν(z) = zJν−1(z).

2.3. BASIC PROPERTIES OF BESSEL FUNCTIONS 13

Similarly,

(2.2)d

dzz−nuJν(z) = −z−νJν+1(z).

and this yields

(2.3) zJ ′ν(z)− νJν(z) = −zJν+1(z).

Substracting and adding the above recurrence formulae yield

Jν−1(z) + Jν+1(z) =2ν

zJν(z)(2.4)

Jν−1(z)− Jν+1(z) = 2J ′ν(z).(2.5)

2.3.3 Generating Function for Jn

Jacobi in 1836 gave

(2.6) e12z(t− 1

t) =

+∞∑k=−∞

tkJk(z).

Many of the forumulae derived above can be obtained from this expression.

(2.7) e12z(t− 1

t) =

+∞∑k=−∞

ck(z)tk

for 0 < |t| <∞. We multiply the power series

(2.8) ezt2 = 1 +

(z/2)

1!t+

(z/2)2

1!t2 + · · ·

and

(2.9) e−zt2 = 1− (z/2)

1!t+

(z/2)2

1!t2 − · · ·

Multiplying the two series and comparing the coefficients of tk yield

cn(z) = Jn(z), n = 0, 1, · · ·(2.10)

cn(z) = (−1)nJ−n(z), n = −1, −2, · · · .(2.11)

Thus

(2.12) e12z(t− 1

t) = J0(z) +

+∞∑k=1

Jk[tk + (−1)kt−k].


2.3.4 Lommel’s Polynomials

Iterating the recurrence formula

(2.13) Jν+1(z) =2ν

zJν(z)− Jν−1

with respect to ν a number of times give

(2.14) Jν+k(z) = P (1/z)Jν(z)−Q(1/z)Jν−1.

Lommel (1871) [6, pp. 294–295] found that

(2.15) Jν+k(z) = Rk,ν(z)Jν(z)−Rk−1,ν+1Jν−1.

2.3.5 Bessel Functions of half-integer Orders

One can check that

(2.16) J− 12(z) =

( 2

πz

) 12

cos z, J 12(z) =

( 2

πz

) 12

sin z.

Moreover,

(2.17) Jn+ 12(z) = (−1)n

√2

πzn+ 1

2

( d

z dz

)n sin z

z, n = 0, 1, 2, · · · .

Thus applying a recurrence formula and using the Lommel polynomials yield

(2.18) Jn+ 12(z) = Rn,ν(z)J 1

2(z)−Rn−1,ν+1J− 1

2(z)

That is, we have

(2.19) Jn+ 12(z) = Rn,ν(z)

( 2

πz

) 12

sin z −Rn−1,ν+1

( 2

πz

) 12

sin z.

2.3.6 Formulae for Lommel’s polynomials

For each fixed ν, the Lommel polynomials are given by

(2.20) Rn ν(z) =

[n/2]∑k=0

(−1)k(n− k)!(ν)n−kk!(n− 2k)!(ν)k

(2

z

)n−2k

where the [x] means the largest integer not exceeding x.

Lommel is a German mathematician who made a major contribution to Besselfunctions around 1870s.

2.3. BASIC PROPERTIES OF BESSEL FUNCTIONS 15

2.3.7 “Pythagoras’ Theorem” for Bessel Function

These Lommel polynomials have remarkable properties. Since

(2.21) J− 12(z) =

( 2

πz

) 12

cos z, J 12(z) =

( 2

πz

) 12

sin z

and sin2 x+ cos2 x = 1; we now have

(2.22) J2n+ 1

2(z) + J2

−n− 12(z) = 2(−1)n

R2n, 12−n(z)

πz.

That is, we have

(2.23) J2n+ 1

2(z) + J2

−n− 12(z) =

2

πz

n∑k=0

(2z)2n−2k(2n− k)!(2n− 2k)!

[(n− k)!]2k!.

A few special cases are

1. J212

(z) + J2− 1

2

(z) =2

πz;

2. J232

(z) + J2− 3

2

(z) =2

πz

(1 +

1

z2

);

3. J252

(z) + J2− 5

2

(z) =2

πz

(1 +

3

z2+

9

z4

);

4. J272

(z) + J2− 7

2

(z) =2

πz

(1 +

6

z2+

45

z4+

225

z6

)

2.3.8 Orthogonality of the Lommel Polynomials

Let us set

(2.24) hn, ν(z) = Rn, ν(1/z) =

[n/2]∑k=0

(−1)k(n− k)!(ν)n−kk!(n− 2k)!(ν)k

(z2

)n−2k

,

then the set {hn ν(z) is called the modified Lommel polynomials. Since the Besselfunctions Jν(z) satisfies a three-term recurrence relation, so the Lommel polynomialsinherit this property:

(2.25) 2z(n+ ν)hn ν(z) = hn+1, ν(z) + hn−1, ν(z)


with initial conditions

(2.26) h0, ν(z) = 1, h1, ν(z) = 2νz.

If one start with a different set of initial conditions

(2.27) h∗0, ν(z) = 0, h∗1, ν(z) = 2ν,

then the sequence {h∗n, ν(z)} generated by the (2.25) is called the associated Lom-mel polynomials. It is known that a three-term recurrence relation for polynomialswith the coefficients of as in (2.25) will generate a set of orthogonal polynomialson (−∞, +∞). That is, there is a probability measure α onb (−∞, +∞) with∫ +∞−∞ dα = 1

Theorem 2.3.1 (A. A. Markov, 1895). Suppose the set of {pn(z)}of orthogonalpolynomials with its measure α supported on a bounded internal [a, b], then

(2.28) limn→∞

p∗n ν(z)

pn ν(z)=

∫ b

a

dα(t)

z − t

holds uniformly for z 6∈ [a, b].

Since we know

Theorem 2.3.2 (Hurwitz). The limit

(2.29) limn→∞

(z/2)ν+nRn, ν+1(z)

Γ(n+ ν + 1)= Jν(z),

holds uniformly on compact subsets of C.

So we have

Theorem 2.3.3. For ν > 0, the polynomials {hn ν(z) are orthogonal with respect toa discrete measure αν normalized to have

∫ +∞−∞ dα(t) = 1, and

(2.30)

∫R

dα(t)

z − t= 2ν

Jν(1/z)

Jν−1(1/z).

One can use this formula to recover the measure α(t). Thus the Lommel “poly-nomials” was found to have very distinct properties to be complete in L2(−1,+1).

2.4. INTEGRAL FORMAULAE 17

2.4 Integral formaulae

(2.1)

Jν(z) =(z/2)ν

Γ(12)Γ(ν + 1

2)

∫ 1

−1

(1− t2)ν−1/2 cos zt dt, <(ν) > −1/2, | arg z| < π.

Or equivalently,(2.2)

Jν(z) =(z/2)ν

Γ(12)Γ(ν + 1

2)

∫ π

0

cos(z cos θ) sin2ν θ dθ, <(ν) > −1/2, | arg z| < π,

where t = cos θ.

Writing H(1)ν (z) = Jν(z) + iYν(z). Then

Theorem 2.4.1 (Hankel 1869).

(2.3) H(1)ν (z) =

( 2

πz

) 12 ei[z−νπ/2−π/4]

Γ(ν + 1/2)

∫ ∞·exp(iβ)

0

e−uuν−12

(1 +

iu

2z

)ν− 12du,

where |β| < π/2.

2.5 Asymptotic Behaviours of Bessel Functions

Expanding the integrand of Henkel’s contour integral by binomial expansion andafter some careful analysis, we have

Theorem 2.5.1. For −π < arg z < 2π,

(2.1) H(1)ν (x) =

(2

πx

) 12

ei(x−12νπ− 1

4π)

[ p−1∑m=0

(12− ν)m(1

2+ ν)m

(2ix)mm!+R(1)

p (x)

],

and for −2π < arg x < π,

(2.2) H(2)ν (x) =

(2

πx

) 12

e−i(x−12νπ− 1

4π)

[ p−1∑m=0

(12− ν)m(1

2+ ν)m

(2ix)mm!+R(2)

p (x)

],

where

(2.3) R(1)p (x) = O(x−p) and R(2)

p (x) = O(x−p),

as x → +∞, uniformly in −π + δ < arg x < 2π − δ and −2π + δ < arg x < π − δrespectively.


The two expansions are valid simultaneously in −π < arg x < π.

We thus have Jν(z) = 12(H

(1)ν (z) +H

(2)ν (z)). So

Theorem 2.5.2. For | arg z| < π, we have

Jν(z) ∼(

2

πz

) 12[

cos(z − 1

2νπ − 1

4π) ∞∑k=0

(−1)k(ν, 2k)

(2z)2k

− sin(z − 1

2νπ − 1

4π) ∞∑k=0

(−1)k(ν, 2k + 1)

(2z)2k+1

]

2.5.1 Addition formulae

Schlafli (1871) derived

(2.4) Jν(z + t) =∞∑

k=−∞

Jν−k(t)Jk(z).

Theorem 2.5.3 (Neumann (1867)). Let z, Z, R forms a triangle and let φ be theangle opposite to the side R, then

(2.5) J0

{√(Z2 + z2 − 2Zz cosφ)

}=∞∑k=0

εkJk(Z)Jk(z) cos kφ,

where ε0 = 1, εk = 2 for k ≥ 1.

We note that Z, z can assume complex values. There are generalizations toν 6= 0.

2.6 Fourier-Bessel Series

Theorem 2.6.1. Suppose the real function f(r) is piecewise continuous in (0, a)and of bounded variation in every [r1, r2] ⊂ (0, a). Then if

(2.1)

∫ a

0

√r|f(r)| dr <∞,

2.6. FOURIER-BESSEL SERIES 19

then the Fouier-Bessel series

(2.2) f(r) =∞∑k=1

ckJν(xνk

r

a

)converges to f(r) at every continuity point of f(r) and to

(2.3)1

2[f(r + 0) + f(r − 0)]

at every discontinuity point of f(r). Here

(2.4) ck =2

J2ν (λk)

∫ 1

0

xf(x)Jν(λkx) dx

is the Fourier-Bessel coefficients of f .

The xνk are the zero of Jν(x) k = 0, 1, · · · . The discussion of the orthogonalityof the Bessel functions is omitted. You may consult G. P. Tolstov’s “ Fourier Series”,Dover 1962.

The situation here is very much like the classical Fourier series in terms of sineand cosine functions.


Chapter 3

An New Perspective

3.1 Hypergeometric Equations

We shall base our consideration of functions on the finite complex plane C in thischapter. Let p(z) and q(z) be meromorphic functions defined in C. A point z0

belongs to C is called a regular singular point of the second order differential equation

(3.1)d2f(z)

dz2+ p(z)

df(z)

dz+ q(z)f(z) = 0

if

(3.2) limz→z0

(z − z0)p(z) = A, limz→z0

(z − z0)2q(z) = B

both exist and being finite. We start by recalling that any second order linearequation with three regular singular points in the C can always be transformed ([2,pp. 73–75]) to the hypergeometric differential equation (due to Euler, Gaussand Riemann)

(3.3) z(1− z)d2y

dz2+ [c− (a+ b+ 1)z]

dy

dz− ab y = 0

which has regular singular points at 0, 1, ∞.

It is known that the hypergeometric equation has a power series solution

(3.4) 2F1

(a, bc

∣∣∣∣ z) = 2F1 [a, b; c; z] =+∞∑k=0

(a)k(b)k(c)k k!

zk

21

22 CHAPTER 3. AN NEW PERSPECTIVE

which converges in |z| < 1. It is called the Gauss hypermetric series whichdepends on three parameters a, b, c. Here the notation is the standard Pochhammernotation (1891)

(a)k = a · (a+ 1) · (z + 2) · · · (a+ k − 1), (ν)0 = 1

for each integer k.

Theorem 3.1.1 (Euler 1769). Suppose <(c) > <(b) > 0, then we have

(3.5) 2F1

(a, bc

∣∣∣∣ z) =Γ(c)

Γ(b)Γ(c− b)

∫ 1

0

tb−1(1− t)c−b−1(1− xt)−a dt,

defined in the cut-plane C\[1, +∞), arg t = arg(1 − t) = 0, and that (1 − xt)−a

takes values in the principal branch (−π, π).

Thus Euler’s integal representation of 2F1 thus continues the function analyt-ically into the cut-plane.

Special cases of (3.4) include: ...

In 1812 Gauss gave a complete set of 15 contiguous relations ...

We note that one can write familiar functions in terms of pFq:

1. ex = 0F0

(−−

∣∣∣∣x);

2. (1− x)−a = 1F0

(a−

∣∣∣∣x);

3. log(1 + z) = x 2F1

(1, 1

2

∣∣∣∣− x);

4. cosx = 0F1

(−

1/2

∣∣∣∣− x2

4

);

5. sin−1 x = x 2F1

(1/2, 1/2

3/2

∣∣∣∣x2

).

3.2. CONFLUENT HYPERGEOMETRIC EQUATIONS 23

3.2 Confluent Hypergeometric Equations

Writing the (3.1) in the variable x = z/b results in the equation

(3.1) x(1− x/b)d2y

dx2+ [c− (a+ b+ 1)x/b]

dy

dx− a y = 0

which has regular singular points at 0, b, ∞. Letting b tends to ∞ results in

(3.2) xd2y

dx2+ (c− x)

dy

dx− a y = 0

which is called the confluent hypergeometric equation. But the equation hasa regular singular point at 0 and an irregular singular point at ∞.

Substituting z/b in (3.4) and letting b to infinity results in a series

(3.3) Φ(a, c; z) := 1F1 [a; c; z] =+∞∑k=0

(a)k(c)k k!

zk.

This series is called a Kummer series. We note that the Gauss hypergeometricseries has radius of convergence 1, and after the above substitution of z/b results ina series that has radius of convergence equal to |b|. So the Kummer series has radiusof convergence∞ after taking the limit b→∞. We call all solutions to the equation(3.2) confluent hypergeometric functions because they are obtained from theconfluence of singularities of the hypergeometric equations. Further details can befound in [4], better known as the Bateman project volumn one.

3.3 A Definition of Bessel Functions

We define the Bessel function of first kind of order ν to be the complex functionrepresented by the power series

(3.1) Jν(z) =+∞∑k=0

(−1)k(12z)ν+2k

Γ(ν + k + 1) k!= zν

+∞∑k=0

(−1)k(12)ν+2k

Γ(ν + k + 1) k!z2k.

Here ν is an arbitrary complex constant and the notation Γ(ν) is the Euler Gammafunction defined by

(3.2) Γ(z) =

∫ ∞0

tz−1e−t dt, <(z) > 0.


It can be analytically continued onto the whole complex plane C, and has simplepoles at the negative integers including 0. So it is a meromorphic function. Inparticular, Γ(k + 1) = k! = k(k − 1) · · · 2 · 1. So the Euler-Gamma function is ageneralization of the factorial notation k!.

Let us set a = ν + 12, c = 2ν + 1 and replace z by 2iz in the Kummer series.

That is,

Φ(ν + 1/2, 2ν + 1; 2iz) =∞∑k=0

(ν + 12)k

(2ν + 1)k k!(2iz)k

= 1F2

(ν + 1

2

2ν + 1

∣∣∣∣ 2iz)= eiz0F1

(−

ν + 1

∣∣∣∣− (z2)2)

= eiz∞∑k=0

(−1)k

(ν + 1)k k!

z2k

22k

= eiz∞∑k=0

(−1)kΓ(ν + 1)

Γ(ν + k + 1) k!

(z2

)2k= eizΓ(ν + 1)

(z2

)−ν · (z2

)ν ∞∑k=0

(−1)k

Γ(ν + k + 1) k!

(z2

)2k= eizΓ(ν + 1)

(z2

)−ν · Jν(z),

where we have applied Kummer’s second transformation

(3.3) 1F1

(a2a

∣∣∣∣ 4x) = e2x 0F1

(−

a+ 12

∣∣∣∣x2

)in the third step above, and the identity (ν)k = Γ(ν + k)/Γ(ν) in the fourth step.

Since the confluent hypergeometric function Φ(ν + 1/2, 2ν + 1; 2iz) satisfiesthe hypergeometric equation

(3.4) zd2y

dz2+ (c− z)

dy

dz− a y = 0

with a = ν+ 1/2 and c = 2ν+ 1. Substituting eizΓ(ν+ 1)(z2

)−ν ·Jν(z) into the (3.4)and simplfying lead to the equation

(3.5) z2d2y

dz2+ x

dy

dz+ (z2 − ν2)y = 0.

3.3. A DEFINITION OF BESSEL FUNCTIONS 25

The above derivation shows that the Bessel function of the first kind with orderν is a special case of the confluent hypergeometric functions with specifically chosenparameters.


Chapter 4

Physical Applications of BesselFunctions

We consider radial vibration of circular membrane. We assume that an elasticcircular membrane (radius `) can vibrate and that the material has a uniform density.Let u(x, y, t) denote the displacement of the membrane at time t from its equilibriumposition. We use polar coordinate in the xy−plane by the change of variables:

(4.1) x = r cos θ, y = r sin θ.

Then the corresponding equation

(4.2)∂2u

∂t2= c2

(∂2u

∂x2+∂2u

∂y2

)can be transformed to the form:

(4.3)∂2u

∂t2= c2

(∂2u

∂r2+

1

r

∂2u

∂r+

1

r2

∂2u

∂θ2

).

Since the membrane has uniform density, so u = u(r, t) that is, it is independent ofthe θ. Thus we have

(4.4)∂2u

∂t2= c2

(∂2u

∂r2+

1

r

∂2u

∂r

).

The boundary condition is u(`, t) = 0, and the initial conditions take the form

(4.5) u(r, t) = f(r),∂u(r, 0)

∂t= g(r).

Separation of variables method yields

(4.6) u(r, t) = R(r)T (t),

27

28 CHAPTER 4. PHYSICAL APPLICATIONS OF BESSEL FUNCTIONS

which satisfies the boundary condition u(`, t) = 0. Thus

(4.7) RT ′′ = c2(R′′T +

1

rR′T

).

Hence

(4.8)R′′ + (1/r)R′

R+

T ′′

c2T; = −λ2,

where the λ is a constant. Thus

R′′ +1

rR′ + λ2R = 0(4.9)

T ′′ + c2λ2T = 0.(4.10)

We notice that the first equation is Bessel equation with ν = 0. Thus its generalsolution is given by

(4.11) R(r) = C1J0(λr) + C2Y0(λr).

. Since Y0 is unbounded when r = 0, so C2 = 0. Thus the boundary conditionimplies that

(4.12) J0(λ`) = 0,

implying that µ = λ` is a zero of J0(µ). Setting C1 = 1, we obtain for each integern = 1, 2, 3, · · · ,

(4.13) Rn(r) = J0(λnr) = J0(µn`r),

where µn = λn` is the n−th zero of J0(µ). Thus we have

(4.14) un(r, t) = (An cos cλnt+Bn sin cλnt) · J0(λnr),

for n = 1, 2, 3, · · · . Thus the general solution is given by

(4.15)∞∑n=1

un(r, t) =∞∑n=1

(An cos cλnt+Bn sin cλnt) · J0(λnr),

and by

(4.16) f(r) = u(r, 0) =∞∑n=1

An · J0(λnr),

29

and

(4.17) g(r) =∂u(r, 0)

∂t

∣∣∣t=0

=∞∑n=1

Bncλn · J0(λnr).

Fourier-Bessel Series theory implies that

(4.18) An =2

`2J21 (µn)

∫ `

0

rf(r)J0(λnr) dr,

(4.19) Bn =2

cλn`2J21 (µn)

∫ `

0

rg(r)J0(λnr) dr.

30 CHAPTER 4. PHYSICAL APPLICATIONS OF BESSEL FUNCTIONS

Chapter 5

Separation of Variables ofHelmholtz Equations

The Helmholtz equation named after the German physicst Hermann von Helmholtzrefers to second order (elliptic) partial differential equations of the form:

(5.1) (∆2 + k2)Φ = 0,

where k is a constant. If k = 0, then it reduces to the Laplace equations.

In this discussion, we shall restrict ourselves in the Euclidean space R3. Oneof the most powerful theories developed in solving linear PDEs is the the methodof separation of variables. For example, the wave equation

(5.2)(

∆2 − 1

c2∂2

∂t2

)Ψ(r, t) = 0,

can be solved by assuming Ψ(r, t) = Φ(r) · T (t) where T (t) = eiωt. This yields

(5.3)(

∆2 − ω2

c2

)Φ(r) = 0,

which is a Helmholtz equation. The questions now is under what 3−dimensionalcoordinate system (u1, u2, u3) do we have a solution that is in the separation ofvariables form

(5.4) Φ(r) = Φ1(u1) · Φ2(u2) · Φ3(u3) ?

Eisenhart, L. P. (”Separable Systems of Stackel.” Ann. Math. 35, 284-305,1934) determines via a certain Stackel determinant is fulfilled (see e.g. Morse,P. M. and Feshbach, H. “Methods of Theoretical Physics, Part I”. New York:McGraw-Hill, pp. 125–126, 271, and 509–510, 1953).

31

32CHAPTER 5. SEPARATION OF VARIABLES OF HELMHOLTZ EQUATIONS

Theorem 5.0.1 (Eisenhart 1934). There are a total of eleven curvilinear coordi-nate systems in which the Helmholtz equation separates.

Each of the curvilinear coordinate is characterized by quadrics. That is,surfaces defined by

(5.5) Ax2 +By2 + Cz2 +Dxy + Exz + Fyz +Gx+Hy + lz + J = 0.

One can visit http://en.wikipedia.org/wiki/Quadric for some of the quadricsurfaces. Curvilinear coordinate systems are formed by putting relevant orthogonalquadric surfaces. Wikipedia contains quite a few of these pictures. We list the elevencoordinate systems here:

33

Co-ordinate SystemsName of system Transformation formulae Degenerate Surfaces(1) Cartesian x = x, y = y, z = z

(2) Cylindricalx = ρ cosφ, y = ρ sinφ, z = zρ ≥ 0, −π < φ ≤ π

(3) Spherical polarx = r sin θ cosφ, y = r sin θ sinφ,z = r cos θr ≥ 0, 0 ≤ θ ≤ π, −π ≤ φ ≤ π

(4)Paraboliccylinder

x = u2 − v2, y = 2uv,z = zu ≥ 0, −∞ < v < +∞

Half-plane

(5)Ellipticcylinder

x = f cosh ξ cos η, y = f sinh ξ sin η,z = zξ ≥ 0, −∞ < η < +∞

Infinite strip;Plane withstraight aperture

(6)Rotationparaboloidal

x = 2uv cosφ, 2uv sinφ,z = u2 − v2

u, v ≥ 0, −π < φ < πHalf-line

(7)Prolatespheroidal

x = ` sinhu sin v cosφ,y = ` sinhu sin v sinφ,z = ` coshu cos v,u ≥ 0, 0 ≤ v ≤ π, −π < φ ≤ π

Finite line ;segmentTwo half-lines

(8)Oblatespheroidal

x = ` coshu sin v cosφ,y = ` coshu sin v sinφ,z = ` sinhu cos v,u ≥ 0, 0 ≤ v ≤ π, −π < φ ≤ π

Circular plate (disc);Plane withcircular aperture

(9) Paraboloidal

x = 12`(cosh 2α + 2 cos 2β − cosh 2γ,

y = 2` coshα cos β sinh γ,z = 2` sinhα sin β cosh γ,α, γ ≥ 0, −π < β ≤ π

Parabolic plate;Plane withparabolic aperture

(10) Elliptic conal

x = krsnαsn β;y = (ik/k′)rcnαcn β,z = (1/k′)rdnαdn β;r ≥ 0, −2K < α ≤ 2K,β = K + iu, 0 ≤ u ≤ 2K ′

Plane sector;Including quarterplane

(11) Ellipsoidal

x = k2`snαsn βsn γ,y = (−K2`/k′)cnαcn βcn γ,z = (i`/k′)dnαdn βcn γ;α, β as in (10), γ = iK ′ + w, 0 < w ≤ K

Elliptic plate;Plane withelliptic aperture


Laplace & HelmholtzCoordinate system Laplace Equation Helmholtz equation

(1) Cartesian (Trivial) (Trivial)

(2) Cylinderical (Trivial) Bessel

(3) Spherical polar Associated Legender Associated Legender

(4) Parabolic cylinder (Trivial) Weber

(5) Elliptic cylinder (Trivial) Mathieu

(6) Rotation-paraboloidal Bessel Confluent hypergeometric

(7) Prolate spheroidal Associated Legender Spheroidal wave

(8) Prolate spheroidal Associated Legender Spheroidal wave

(9) Paraboloidal Mathieu Whittaker-Hill

(10) Elliptic conal Lame Spherical Bessel, Lame

(11) Ellipsoidal Lame Ellipsoidal

35

1. Associated Legender:

(5.6) (1− x2)d2y

dx2− 2x

dy

dx+{n(n+ 1)− m2

(1− x2)

}y = 0

2. Bessel:

(5.7) x2 d2y

dx2+ x

dy

dx+ (x2 − n2)y = 0

3. Spherical Bessel:

(5.8) x2 d2y

dx2+ 2x

dy

dx+ (x2 − n(n+ 1))y = 0

4. Weber:

(5.9)d2y

dx2+ (λ− 1

4x2)y = 0

5. Confluent hypergeometric:

(5.10) xd2y

dx2+ (γ − x)

dy

dx− αy = 0

6. Mathieu:

(5.11)d2y

dx2+ (λ− 2q cos 2x)y = 0

7. Spheroidal wave:

(5.12) (1− x2)d2y

dx2− 2x

dy

dx+{λ− µ2

(1− x2)+ γ2(1− x2)

}y = 0

8. Lame:

(5.13)d2y

dx2+ (h− n(n+ 1)k2sn 2x)y = 0

9. Whittaker-Hill:

(5.14)d2y

dx2+ (a+ b cos 2x+ c cos 4x)y = 0

10. Ellipsoidal wave:

(5.15)d2y

dx2+ (a+ bk2sn 2x+ qk4sn 4x)y = 0

Remark 5.0.1. The spheroidal wave and the Whittaker-Hill do not belong to thehypergeometric equation regime, but to the Heun equation regime (which has fourregular singular points).


5.1 What is not known?

It is generally regarded that the Bessel functions, Weber functions, Legendre func-tions are better understood, but the remaining equations/functions are not so wellunderstood.

1. Bessel functions. OK! Still some unknowns.

2. Confluent hypergeometric equations/functions. NOT OK.

3. Spheroidal wave, Mathieu, Lame, Whittaker-Hill, Ellipsoidal wave are poorlyunderstood. Some of them are relatd to the Heun equation which has fourregular singular points. Its research has barely started despite the fact thatit has been around since 1910.

4. Mathematicans are separating variables of Laplace/Helmholtz equations, butin more complicated setting (such as in Riemannian spaces, etc)

Bibliography

[1] Milton Abramowitz and Irene A. Stegun, editors. Handbook of mathematicalfunctions, with formulas, graphs, and mathematical tables, volume 55 of NationalBureau of Standards Applied Mathematics Series. Superintendent of Documents,US Government Printing Office, Washington, DC, 1965. Third printing, withcorrections.

[2] G. E. Andrews, R. A. Askey, and R. Roy. Special Functions. Cambridge Uni-versity Press, Cambridge, 1999.

[3] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcen-dental Functions, volume 2. McGraw-Hill, New York, 1953.

[4] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcen-dental Functions, volume 1. McGraw-Hill, New York, 1953.

[5] T. J. Rivlin and M. W. Wilson. An optimal property of Chebyshev expansions.J. Approximation Theory, 2:312–317, 1969.

[6] G. N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge Uni-versity Press, Cambridge, second edition, 1944.

37