14. Bessel Functions

1. Bessel Functions of the 1st Kind, J (x)

2. Orthogonality

3. Neumann Functions, Bessel Functions of the 2nd Kind

4. Hankel Functions, I (x) and K (x)

5. Asymptotic Expansions

6. Spherical Bessel Functions

Defining Properties of Special Functions

1. Differential eq.

2. Series form / Generating

function.

3. Recurrence relations.

4. Integral representation.

Basic Properties :

1. Orthonormality.

2. Asymptotic form.

Ref :1.M.Abramowitz & I.A.Stegun, Handbook of

Mathematical Functions, Dover Publ. (1970)

http://people.math.sfu.ca/~cbm/aands/abramowitz_

and_stegun.pdf.2.NIST Digital Library of Mathematical Functions:

http://dlmf.nist.gov/

Usage of Bessel Functions

Solutions to equations involving the Laplacian, 2 , in

circular cylindrical coordinates : Bessel / Modified Bessel functions

or spherical coordinates : Spherical Bessel functions

1. Bessel Functions of the 1st Kind, J (x)

2 2 2 0x J xJ x J Bessel functions are Frobenius solutions of the Bessel ODE

1st kind Jn (x) : n = 0, 1, 2, 3, … regular at x = 0.

2

00

2 !! ! 2

j j

j

xJ x a

j j

for 1, 2, 3, …

(eq.7.48)

0

1

2 !na

n

Mathematica

cf. gen. func.

Periodic with amp x 1/2 as x .

( in eq.7.48 )

Generating Function for Integral Order

Generating function : 1, exp

2

xg x t t

t

n

nn

J x t

0

1

2 !na

n

/2 /2, xt x tg x t e e 0 0! 2 ! 2

r sr ss

r s

t x t x

r s

2

max 0, ! ! 2

s n sn

n s n

xt

n s s

0 0 ! ! 2

r sr ss

r s

t x

r s

n r s

For n 0

2

0 ! ! 2

s n s

ns

xJ x

n s s

n = m < 0 :

2

! ! 2

s m s

ms m

xJ x

m s s

2

0 ! ! 2

t m m t

t

x

t t m

m

m mJ x J x

2

0 1 ! 2

s s

s

xJ x

s s

1, 2, 3,

0max 0,

0

n ss n

s

Generalize:

Recurrence /2 /2, xt x t nn

n

g x t e e J x t

2

1, , 1

2t

xg x t g x t

t

2

2n n

nn

xJ x t t

1 1, ,

2x g x t g x t tt

nn

n

J x t

1nn

n

J x n t

22n

n nn

xt J x J x

2 1

2 1n n n

nJ x J x J x

x

1 1 nn

n

J x n t

1 1

2n n n

nJ x J x J x

x

1 11

2n n

nn

J x t t

1 1

1

2n

n nn

t J x J x

1 1 2n n nJ x J x J x

1 0J x J x n

n nJ x J x

Ex. 14.1.4

1

1

1 1

1

n nn n

n nn n

n n n

dx J x x J x

d x

dx J x x J x

d x

nJ x J x J x

x

Bessel’s Differential Equation 2 2 2 0x J xJ x J

1 1

2n n n

nJ x J x J x

x

1 1 2n n nJ x J x J x

2

0 1 ! 2

s s

s

xJ x

s s

Any set of functions Z (x) satisfying the

recursions must also satisfy the ODE,

though not necessarily the series expansion.

Proof :

1 1

1

2Z x Z x Z x

2

21 12

xx Z x Z x Z x

2 2

2

1 1 1 1 1 1

1

2

x Z x xZ x Z x

xZ x Z x Z x Z x Z x Z x

x x

1 12

xxZ x Z x Z x

21 12

xZ x Z x Z x

2 2

2

1 1 1 1 1 1

1

2

x Z x xZ x Z x

xZ x Z x Z x Z x Z x Z x

x x

1 1

1n n n

nJ x J x J x

x 1 1

1Z x Z x Z x

x

2

1 1 1 1

1 1

2

xZ x Z x Z x Z x

x x

2 2 2x Z x xZ x Z x x Z x QED

2

2 2 2 22

0d d

Z k Z k k Z kd d

x k

Integral Representation :Integral Order

1exp

2n

nn

xt J x t

t

/2 1/1

1

x t tm n

mnC Cm

ed t J x d t t

t

C encloses t = 0.n = integers

C = unit circle centered at origin :

2 ni J x

it e 1i n

n

d ti e d

t

1 i it e et

2

sin

0

2 in i xni J x i d e

2 sini

2

0

cos sin sin sini d x n i x n

Re : 2

0

1cos sin

2nJ x d x n

2

0

sin sin 0d x n

0

1cos sinnJ x d x n

1cos sin

2d x n

Im :

n = integers

2sin

0

0

0

1

2

1cos sin

i xJ x d e

d x

2

cos sinnJ x d x n

0

cos sind x n n

0

cos sinn

d x n

0

cos sind x n

n = integers

n

n nJ x J x

3 /2

cos0

/2

1

2i xJ x d e

2

cos

0

1

2i xd e

0

cos sind x n

Zeros of Bessel Functions

nk : kth zero of Jn(x)

nk : kth zero of Jn(x) Mathematica

kth zero of J0(x) = kth zero of J1(x)

kth zero of Jn(x) ~ kth zero of Jn-1(x)

Example 14.1.1. Fraunhofer Diffraction, Circular Aperture

Fraunhofer diffraction (far field) for incident plane wave, circular aperture :

Kirchhoff's diffraction formula (scalar amplitude of field) :

1

4

i k i k

S

e ed

r r r r

r r r Sr r r r

2cos

0 0

~a

i b rd r r d e

sinb k

2k

sin , 0, cosr r cos , sin , 0a r

2 2 2 2 2sin cos sin cosr a a r r r

sin cosr a

Mathematica

2cos

0 0

~a

i b rd r r d e

sinb k Primes on variables dropped for clarity.

0

0

~ 2a

d r r J br 2

cos0

0

1

2i xJ x d e

1n n

n n

dx J x x J x

d x 120

2~

bad

d x x J xb d x

1

2 aJ ba

b

2

21

2~

aJ ba

b

Intensity:

2

1

2sin

sin

aJ ka

k

Mathematica

1st min: 11sin 3.8317ka

14

Example 14.1.2. Cylindrical Resonant Cavity

Wave equation in vacuum :2

22 2

10

c t

F orF E B

Circular cylindrical cavity, axis along z-axis :

/ /0, 0z SB E

2 22

2 2 2

1 1

z

TM mode :

, ,zE z R Z z

i te F2

22

0c

F

2 2 2

2 2 2 2

1 10

d d R d d Z

R d d d Z d z c

/ / / /

0,

0 0

0

0

S

z

z h

E

z

E

E

// means tangent to wall S

cosp

pZ z z

h

kc

0,1, 2, 3,p

22

20

dl Z

d z

with

22

20

dm

d

2 2 2 2 0d d R

k l m Rd d

2 22

2 2 2

1 10

d d R d d Zk

R d d d Z d z

0,

0z

z h

E

z

p

lh

0 2z zE E

sin cosm A m B m 0,1, 2, 3,m

0z aE

m j p m m jR J

a

mj = jth zero of Jm(x) .

222 m jp

kh a

2

2 2 2 22

0d d

Z k Z k k Z kd d

2 2

mnm j p

pc

a h

resonant frequency

Caution: are linearly independent.

Bessel Functions of Nonintegral Order

2

0 ! ! 2

j j

j

xJ x

j j

Formally, gives only Jn of integral order.

with

1exp

2n

nn

xt J x t

t

However, the series expansion can be extended to J of nonintegral order :

for 1, 2, 3, …

n

n nJ x J x

&J x J x

2

0 ! ! 2

j n j

nj

xJ x

n j j

Strategy for proving

1. Show F satisfies Bessel eq. for J .

2. Show for x 0.

For nonintegral , is multi-valued.

Possible candidate for is

Schlaefli Integral

/2 1/

1

1

2

x t t

n nC

eJ x d t

i t

1

1

t

C encloses t = 0.n = integers

1

/2 1/

1

1

2

x t t

C

eF x d t

i t

J x

/2 1/

1Relim 0

x t t

t

e

t

F x J x

F x J x

Mathematica

/2 1/

1

1

2

x t t

C

eF x d t

i t

/2 1/

1

1 1 1

2 2

x t t

C

eF x d t t

i t t

2/2 1/

1

1 1 1

2 4

x t t

C

eF x d t t

i t t

2 2 2 0x J xJ x J

/2 1/

2 2 2 1 1

2 2

end

start

tx t t

t

e xx F xF x F t

i t t

/2 1/ /2 1/

1

1

2

x t t x t td e e xt

d t t t t

2/2 1/ /2 1/ 22

1

1 1 1

2 4 2

x t t x t td e x e x xt t t

d t t t t t t

2/2 1/ 2

2 2 2 21

1 1 1

2 4 2

x t t

C

e x xx F xF x F d t t t

i t t t

Consider any open contour C that doesn’t cross the branch cut

Set :

0/2 1/

0

1 1

2 2

t ix t t

t i

e xRHS t

i t t

0

/2 1/

1

1

2

x t t

C

eF x d t

i t

For 1 0x /2

1

1

2

xt

C

ed t

i t

1

2u xt

11

2 2i u

C

xF x e du e u

i

C = spatial inversion of C , ( same as that for ; B.cut. on +axis ) .

2 111

2 2iix

e ei

1sin 1

2

x

1

1 2

x

QED 1J x

1sin

z zz

Mathematica

/2 1/

2 2 2 1 1

2 2

end

start

tx t t

t

e xx F xF x F t

i t t

For C1 :

this F is a solution of the Bessel eq.

2. Orthogonality

2 2 2 2 0Z k Z k k Z k

2Z k k Z k L2 2

2 2

1d d

d d

Lwhere

i.e., Z (k) is the eigenfunction, with eigenvalue k2 , of the operator L .

( Sturm-Liouville eigenvalue probem )

Helmholtz eq. in cylindrical coordinates

with

2 2 0

P Z z 2 2

22 2

1d d mP k P

d d

mP J k 0P a mnka

mn = nth root of Jm

2 2 2zk l

L is Hermitian, i.e., , if the inner product is defined as

2 2

2 2

1d d

d d

L is not self-adjoint

1

0 0

1exp

pw d

p p

1exp d

2d d

d d

L L is self-adjoint

2J k k J k L 2J k k J k L J is an eigenfunction of with eigenvalue k2.

L L

*

0

d

* * * *d d d d

d d d d

L L

*

* * * d d d d

d d d d

L L

**d d d

d d d

0

a

d J k J k J k J k L L

2 2

0

a

k k d J k J k

a J k a k J ka k J k a J ka d J k

k J kd

L L

2J k k J k L

1

2 2

0

a

a k J k a J ka J k a k J ka k k d J k J k

Orthogonal Sets

orthogonal set

; 0, 1, 2,jJ ja

2 2

0

a

a k J k a J ka J k a k J ka k k d J k J k

Let

jka

ik a

2 2

20

0a

j ii jd J J

a a a

0

a

i j i jd J Ja a

Orthogonality :

Let

jka

ik a

2 2

20

0a

j ii jd J J

a a a

0

a

i j i jd J Ja a

Orthogonality : ; 0, 1, 2,jJ ja

orthogonal set

0i jJ J

Mathematica

1

0

i j i jd J J

1

0

i j i jd J J

Normalization

2 2

0

a

a k J k a J ka J k a k J ka k k d J k J k

2

2 20

lima

k k

k J k a J ka J k a k J kad J k a

k k

2

22

0

1

2

a

i id J a Ja

22 2

limy x

y J y J x J y xJ xa

x y

x ka

y k a

2lim2y x

y J y J x J y xJ xa

y

2

2

2

x J x J x xJ xa

x

2

22

0

1

2

a

i id J a Ja

1 1

1n n n

nJ x J x J x

x 1n n n

nJ x J x J x

x

1 i iJ J

2

22

1

0

1

2

a

i id J a Ja

Mathematica

Similarly :

(see Ex.14.2.2)

2 2 22

20

11

2

a

i ii

d J a Ja

Bessel Series : J ( i / a )

22

1

0

1

2

a

i j i i jd J J a Ja a

For any well-behaved function f () with f (a) 0 :

1

j jj

f c Ja

for any > 1

2

201

2 a

j j

j

c d J faa J

with

Bessel Series J ( i / a )

For any well-behaved function f () with f (a) 0 :

1

j jj

f d Ja

for any > 1

2 2 0212

2

1

a

j j

jj

d d J fa

a J

with

2 2 2

22

0

11

2

a

i ii

d J a Ja

Example 14.2.1. Electrostatic Potential: Hollow Cylinder

Hollow cylinder : axis // z-axis, ends at z = 0, h ; radius = a.

Potentials at boundaries :

00

z aV V

,

z hV f

Electrostatics, no charges : 2 0V

Eigenstates with cylindrical symmetry : , , z P Z z

i mm e

2 2

2 2 2

1 10

d d P d d Z

P d d d Z d z

22

2

dm

d

sinhZ l z

22

2

d Zl Z

d z

0 0Z

22

20

d d P ml P

d d

22

20

d d P ml P

d d

i mm e sinhZ l z

mP J l m m jJa

0a

m jla

sinhi m

m j m m j m j

zJ e

a a

1

, , m j m jm j

V z c

2

0

2i m mm md e

22

1

0

1

2

a

i j i i jd J J a Ja a

2

22 0 0

1,

sinh

ai m

m j m m j

m m j m j

c d e d J fh aa Ja