4. Hankel Functions, H(1) (x) & H

(2) (x)

1H x J x i Y x

Hankel functions of the 1st & 2nd kind :

2H x J x i Y x

c.f. cos sinie i

*1 2H x H x

for x real

1

1 2

xJ x

1 !2 2ln

2

xY x J x

x

For x << 1,

> 0 :

0

2ln

2

xY x J x

10

21 ln

2

xH x i

1 1 ! 2~H x i

x

20

21 ln

2

xH x i

2 1 ! 2~H x i

x

Recurrence Relations

1 1

2sinJ J J J

x

1 1

2sinJ J J J

x

2J Y J Y

x

1 1

2J Y J Y

x

1 1

2J x J x J x

x

1 1 2J x J x J x

2 1 1 21 1

4H H H H

i x

1 11 1

2J H J H

i x

1, 2 1, 2 1, 21 1

2H x H x H x

x

1, 2 1 , 2 1 , 21 1 2H x H x H x

2 21 1

2J H J H

i x

Contour Representations

/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/

1

1

2

x t t

C

eF x d t

i t

The integral representation

is a solution of the Bessel eq. if at end points of

C.

See Schlaefli integral

/2 1/

1

1

2

x t t

C

eF x d t

i t

/2 1/ 1

02

x t te xt

t t

/2 1/

0 or Re

10 0

2

x t t

t t

e xt x

t t

1/lim lim 0a t tb b a t

t tt e t e

1/ /

0 0lim lim 0a t tb b a t

t tt e t e

0a

Mathematica

The integral representation

is a solution of the Bessel eq. for any C with end points t = 0 and Re t = .

/2 1/

1

1

2

x t t

C

eF x d t

i t

Consider

1

/2 1/1

1

1 x t t

C

ef x d t

i t

2

/2 1/2

1

1 x t t

C

ef x d t

i t

1 21

2J x f x f x

If one can prove 1 21

2Y x f x f x

i

then 1f x J x i Y x

2f x J x i Y x

1H x

2H x

Proof of 1 21

2Y x f x f x

i

1 , 2

/2 1/1 , 2

1

1 x t t

C

ef x d t

i t

1 ie

ts s

1 1

1 ie

t s

1 1t s

t s

1

/2 1/1

1

x s si

C

e ef x d s

i s

1ie f x

2

/2 1/2

1

x s si

C

e ef x d s

i s

2ie f x

2

d sd t

s

0~

0

i

i

et s

e

1 21

2J x f x f x

1 21

2i ie f x e f x

1 21

2i iJ x e f x e f x

1 21

2J x f x f x

cos

sin

j x j xY x

1 2 1 2cos cos sin cos sin1

2 sin

f x f x i f x i f x

1 21

2f x f x

i QED

1

/2 1/1

1

1 x t t

C

eH x d t

i t

2

/2 1/2

1

1 x t t

C

eH x d t

i t

i.e.

are saddle points.(To be used in asymptotic expansions.)

t i

5. Modified Bessel Functions, I (x) & K (x)

2 2 2 2 0Z k Z k k Z k Bessel equation :

Z k A J k B Y k 1 2C H k D H k

2 2 2 2 0R k R k k R k Modified Bessel equation :

R k A I k B K k

oscillatory

Modified Bessel functions exponential

k ik Bessel eq. Modified Bessel eq.

are all solutions of the MBE. 1 2, , ,J ik Y ik H ik H ik

I (x)

2

0 1 ! 2

s s

s

xJ x

s s

2

0

1

1 ! 2

s

s

xJ ix i

s s

Modified Bessel functions of the 1st kind :

I x i J i x

/ 2 /2i ie J x e

2

0

1

1 ! 2

s

s

x

s s

I (x) is regular at x = 0 with 1

1 2

xI x

n

n nJ x J x nn nI x i J i x nn

ni J i x nn nni i I x

n nI x I x

Mathematica

Recurrence Relations for I (x)

1 1

2J x J x J x

x

1 1 2J x J x J x

I x i J i x

1 1

2J ix J i x J i x

i x

1 11 1

2i I x i I x i I x

i x

1 1

2I x I x I x

x

1 1 2

d J ixJ ix J ix

d ix

1 1 11 1 2i I x i I x i I x

1 1 2I x I x I x

d I x d J ixi i

d x d ix

2nd Solution K (x)

11

2K x i H ix

Modified Bessel functions of the 2nd kind ( Whitaker functions ) :

1

2i J i x i Y ix

2 sin

I x I x

x

1 1

2K x K x K x

x

1 1 2K x K x K x

Recurrence relations :

0 ln ln 2K x x For x 0 :

12K x x

Ex.14.5.9

Integral Representations

cos

0

1cosx

nI x d e n

0

0

1cosh cosI x d x

Ex.14.5.14

0

0

cos sinhK x d t x t

2

0

cos

1

xtd t

t

0x

Example 14.5.1. A Green’s Function

2 2 2

1 2 1 2 1 2 1 22 2 2 2 21 1 1 1 1 1 1

1 1 1,G z z

z

r r

Green function for the Laplace eq. in cylindrical coordinates :

1 2

1 2

1

2i m

m

e

1 2

1 2

1

2i k z z

z z d k e

1 2

0

1cosd k k z z

Let

1 2

1 2 1 2 1 22

0

1, , , cos

2i m

mm

G d k g k e k z z

r r

1 2

2 2 2

1 22 2 2 21 1 1 1 1 1

2 22

1 2 1 22 2 21 1 1 1

0

1 1,

1 1cos , ,

2i m

mm

Gz

md k e k z z k g k

r r

1 2

1 2 1 22 21

0

1 1cos

2i m

m

d k e k z z

2 2

2 21 1 2 1 22 2

1 1 1 1

1, ,m

mk g k

§10.1 1 2,m m mg k k A I k K k

Ex.14.5.11 1m m m mA I k K k I k K k

1A

1 2

1 2 1 2 1 22

0

1, , , cos

2i m

mm

G d k g k e k z z

r r

5. Asymptotic Expansions

1. Expansion in negative powers [ Stokes’ method (Ex 14.6.10.) ].

Problem : Relation to named functions not known.

2. Steepest descent.

Asymptotic Forms of H

Contour integral representation:

1

1/ / 21

1

1 z z t

C

eH t d z

i z

2

1/ / 22

1

1 z z t

C

eH t d z

i z

2

11 0

2

tw

z

0z i

3

4 2 2or

0

1 3arg

2 2 2w z or

1 /2

tw z z

Method of steepest descent ( §12.7 ) :

0

00

2w z w z i

C

d z g z e g z e ew z

0 0w z

3

tw

z

0 3

tw z

i

/ 2it e 3 / 4

/ 4

0w z i t

1 1 3 / 41 2i t iH t i e ei t

1 1 1 / 41 2i i t iH t i e e ei t

1 1 3 / 41 2i t iH t i e ei t

1 1 1 / 41 2i i t iH t i e e ei t

2 1exp

2 2i t i

t

3 2exp 1 1

2 2i t i

t

1 2

exp 1 12 2

i t it

2 1exp

2 2i t i

t

11

2K x i H ix

1/2 1~ exp

2 2 2K x i xt i

x

2xte

x

Expansion of an Integral Representation for K

1/22

1

11 22

z xzR z d x e x

1, Re 0

2z

Proof : 1. R satisfies the MBE. 2 2 2 0z R zR z R

1/22

1

11 22

z xzR z R z d x x e x

z

1/222

1

11 22

z xzR z R z R z d x x x e x

z z z

1/22 2 2 2 2

1

2 1 11 22

z xzz R zR z R z R d x xz zx e x

Consider

1

2

1/2 1/22 22

12

21 11

z x z x

xd

z e x z z e xd x x

1/22 22 1 1z xz zx xz e x

1/22 2 2 2

11

1 22

z xzz R zR z R z e x

10

2 QED

1/22 2 2 2 2

1

2 1 11 22

z xzz R zR z R z R d x xz zx e x

Proof : 2. R = K for z 0.

1/22

1

11 / 2 2

z xzR z d x e x

12K z z

Let 1t

xz

d td x

z

1 0x t

z tz xe e 2 1 2

t tx

z z

1/2

0

21 / 2 2

ztz e t t

R z d t ez z z

1

2

1/22 1

0

21

1 / 2 2

zte z

d t e tz t

2 1

0

1

1 / 2 2td t e t

z

12

1 / 2 2 z

12 z

211 2 2 1

2zz z z

2 112 2

2zz z z

QED

Proofs 1 & 2 R = K i.e.

1/22

1

11 / 2 2

z xzK z d x e x

1, Re 0

2z

Proof : 3. K (z) decays exponentially for large z.

1/2

0

21 / 2 2

ztz e t t

K z d t ez z z

1/21/2

0

11

1 / 2 2 2z t t

e d t e tz z

1/2

00

12

2 ! 1 / 2rz t r

r

e z d t e tz r r

0

1 / 22

2 ! 1 / 2rz

r

rK z e z

z r r

~2

zez

QED

0

1 / 22

2 ! 1 / 2rz

r

rK z e z

z r r

is a divergent asymptotic series

2

2

11 0R z R z R

z z

z = is an essential singularity

No convergent series solution about z = .

1x

z

2d dx

d z d x

2 23 4

2 22

d d dx x

d z d x d x

4 3 2 21 0x R x x R x x R x

22 2

1 1 10R x R x R x

x x x

2 22 20

1 1lim 0x

xx x

2

1 / 2 1 / 2 3 / 2 1 / 2 1 / 2 3 / 21

2 2 2! 2ze

z z z

Series terminates for 1 3

, ,2 2

0

1 / 22

2 ! 1 / 2rz

r

rK z e z

z r r

2

ze P iz i Q izz

2 2 1

0 0

2 1 / 2 2 3 / 22 2

2 2 ! 2 1 / 2 2 1 ! 2 1 / 2n nz

n n

n ne z z

z n n n n

2

0

2 1 / 22

2 ! 2 1 / 2n

n

nP z iz

n n

2 1

0

2 3 / 22

2 1 ! 2 1 / 2n

n

nQ z i iz

n n

2 2 2 2 2 22 2 2 2 2 2

2 4

3 1 7 5 3 12 2 2 2 2 2

~ 12! 2 4! 2z z

2 2 2 22 2 2 2

3

1 5 3 12 2 2 2

~2 3! 2z z

2

0

2 1 / 22

2 ! 2 1 / 2n n

n

nz

n n

2 1

0

2 3 / 22

2 1 ! 2 1 / 2n n

n

nz

n n

Additional Asymptotic Forms

2

zK z e P iz i Q izz

2

0

2 1 / 22

2 ! 2 1 / 2n n

n

nP z z

n n

2 1

0

2 3 / 22

2 1 ! 2 1 / 2n n

n

nQ z z

n n

Asymptotic forms of other Bessel functions can be expressed in terms of P & Q .

11

2K x i H ix

1 12H x i K ix

1 1/2 /22 i z iH z e P z i Q zz

Analytic continued to all z

*2 1H x H x 1/2 /22 i x ie P x i Q xx

2 1/2 /22 i z iH z e P z i Q zz

Analytic continued to all z :

1 1/2 /22 i z iH z e P z i Q zz

1ReJ x H x 2 1 1cos sin

2 2 2 2P x x Q x x

x

2 1 1cos sin

2 2 2 2J z P z z Q z z

z

1ImY x H x 2 1 1sin cos

2 2 2 2P x x Q x x

x

2 1 1sin cos

2 2 2 2Y z P z z Q z z

z

I z i J i z

2

zeI z P iz i Q iz

z

Properties of the Asymptotic Forms

22

~ 1

1 / 2~

21 3

, ,2 2

Series terminates for

P z

Q zz

All Bessel functions have the asymptotic form

1Z z f z P z i g z Q z

z

, , , cos , or sinz i zf z g z e e z z where

e.g. 1 zI z ez

1 zK z ez

2 1~ cos

2 2J z z

z

2 1~ sin

2 2Y z z

z

good for 222 1 / 2z

2 1~ cos

2 2J z z

z

0

2~ cos

4J z z

z

222 1 / 2z

Mathematica

Example 14.6.1. Cylindrical Traveling Waves

Eg. 14.1.24 : 2-D vibrating circular membrane standing waves

Consider 2-D vibrating circular membrane without boundary

travelinging waves

For large r i k x tU e

Circular symmetry (no dependence ) :

10, i tU r t H kr e diverges at r = 0

6. Spherical Bessel Functions

Z krR kr

kr

2

2 2 22

2 1 0d R d R

r r k r l l Rd r d r

Radial part of the Helmholtz eq. in spherical coordinates

1 1

2

d R d ZZ

d r d r rkr

2 2

2 2 2

1 1 1 1 1

2 2 2 2

d R d Z d Z d ZZ Z

d r d r r d r r r d r rkr

2

2 2

1 1 3

4

d Z d ZZ

d r r d r rkr

222 2 2

2

10

2

d Z d Zr r k r l Z

d r d r

1/2

1/2

l

l

J krZ kr

Y kr

1/2

1/2

l

l

J kr

krR kr

Y kr

kr

Spherical Bessel functions

Definitions

1/22n nj x J xx

Spherical Bessel functions ( integer orders only ) :

1/22n ny x Y xx

1 11/22n nh x H x

x

n nj x i y x

2 21/22n nh x H x

x

n nj x i y x

cos

sin

J x J xY x

1/2 1/2

1/2

cos 1 / 2

sin 1 / 2n n

n

J x n J xY x

n

1

1/2

n

nJ x

1

1/22n

n ny x J xx

1

1

n

n ny x j x

2

0 1 ! 2

s s

s

xJ x

s s

1/22n nj x J x

x

2 1/2

02 3 / 2 ! 2

s n s

ns

xj x

x n s s

2

0 32 1 !! 2!2

s sn

ns

s

x xj x

n n s

3 1 1 3 3

2 2 2 2 2n s n s n s n n

3 3

2 2s

n n

Pochhammer symbol 1 1s

n n n n s where

3 1 1 3 1 1

2 2 2 2 2 2n n n n

1

2 1 !!

2n

n

1

1

n

n ny x j x

2 1/2

02 3 / 2 ! 2

s n s

ns

xj x

x n s s

2 1/21

02 1 / 2 ! 2

s n sn

ns

xy x

x n s s

1 1 1 1 11

2 2 2 2 2n n

1 1 3 1 1

2 2 2 2 2n s n s n s n n

1 1

2 2s

n n

12 1 !!

2 2

n

nn n

1/2 21

0

2 1 !!

2 2 2 1 / 2 1 / 2 ! 2

n sn sn

n ns s

nx xy x

x n s

2

10

2 1 !!

1 / 2 ! 2

s s

n ns s

n xy x

x n s

2

00

1

1 / 2 ! 2

s s

s s

xy x

x s

jn & yn

Mathematica

2

0 32 1 !! 2!2

s sn

ns

s

x xj x

n n s

2

10

2 1 !!

1 / 2 ! 2

s s

n ns s

n xy x

x n s

2

00 3 2!

2

s s

s

s

xj x

s

2 23 3 5 1 12 ! 2 1 2 1

2 2 2 2 2s s

s

s s s s s

3 5 2 1 2 1 2 2 2 4 2s s s s 2 1 !! 2 !!s s

20

0 2 1 !

s

s

s

j x xs

2 1 !s

0

sin xj x

x

2 1

0

sin2 1 !

s

s

s

x xs

2

00

1

1 / 2 ! 2

s s

s s

xy x

x s

2 21 1 3 3 12 ! 2 1 2 1

2 2 2 2 2s s

s

s s s s s

2 1 !! 2 !!s s 2 !s

20

0

1

2 !

s

s

s

y x xx s

0

cos xy x

x

2

0

cos2 !

s

s

s

x xs

1 11/22n n n nh x H x j x i y x

x

2 2

1/22n n n nh x H x j x i y xx

10

1sin cosh x x i x

x

0

sin xj x

x 0

cos xy x

x

i xie

x 2

0

1sin cosh x x i x

x i xi

ex

1 1/2 /22 i z iH z e P z i Q zz

111/2 1/2

1n i xn n nh x i e P x i Q x

x

2

0

2 1 / 22

2 ! 2 1 / 2n n

n

nP z z

n n

2 1

0

2 3 / 22

2 1 ! 2 1 / 2n n

n

nQ z z

n n

2

1/20

2 12

2 ! 2 1s s

ns

n sP z z

s n s

10, 2, 4, ,

!2

! !tt

t t

n ti z

t n t

2 1

1/20

2 22

2 1 ! 2s s

ns

n si Q z i z

s n s

2t s

11, 3, 5, ,

!2

! !tt

t t

n ti z

t n t

2 1t s

11

0

!

! !2

i x tnn

n tt

n te ih x i

x t n tx

1 or 1t n n

2 !!!

2n

nn

1

0

2 2 !!

! 2 2 !!8

i x tnn

tt

n te ii

x t n tx

1 for 1, 2,z z

11

0

!

! !2

i x tnn

n tt

n te ih x i

x t n tx

11 1

i xe ih x

x x

1 21 1 1

1

2j x h x h x

1i xe i

x x

12 2

3 31

i xe ih x i

x x x

*2 11 1h x h x

2

cos sinx x

x x

22 2

3 31

i xe ih x i

x x x

2 2 3

sin 3 cos 3sinx x xj x

x x x

1 21

2n n nj x h x h x

2

sin cosx x

x x

1 21

2n n ny x h x h xi 1 2

1 1 1

1

2y x h x h x

i

2 2 3

cos 3 sin 3cosx x xy x

x x x

For any Bessel functions

F (x) = J (x) , Y (x) , H (1,2)(x) : Recurrence Relations

1 1

2F x F x F x

x

1 1 2F x F x F x

For any spherical Bessel functions

fn (x) = jn (x) , yn(x) , hn(1,2)(x) :

1/22n nf x F xx

1 1

12

2n n n nf x f x f x f xx

1 1

2 1n n n

nf x f x f x

x

1/2

1

2 2n n nf x f x F xx x

1/2

1

2 2n n nF x f x f xx x

1 1

12

2 2 1n n nf x f x f xn

1 11 2 1n n nn f x n f x n f x

1

1

dx F x x F x

d x

dx F x x F x

d x

1/22n nf x F xx

1/2 1/21/2 1/2

n nn n

dx F x x F x

d x

1 11

n nn n

dx f x x f x

d x

1/2 1/21/2 3/2

n nn n

dx F x x F x

d x

1n n

n n

dx f x x f x

d x

Rayleigh Formulas

1 sinn

n nn

d xj x x

x d x x

1 cosn

n nn

d xy x x

x d x x

1 1n i x

n nn

d eh x i x

x d x x

2 1n i x

n nn

d eh x i x

x d x x

Proof is by induction.

Proof of Rayleigh Formula

2

1 sin cos sind x x xx

x d x x x x

For n = 1 : 1j x

Assuming case n to be true,

11 1

1

1 sin

1 sin 1 sin

nn n

n nn nn n

d xx

x d x x

d d x d xx nx

d x x d x x x d x x

1 sinn

n nn

d xj x x

x d x x

n n

nj x j x

x

1 11 2 1n n nn f x n f x n f x

1 1

2 1n n n

nf x f x f x

x

1 1 1 1

1

2 1 2 1 2 1n n n n

n n nj x j x j x j x

n n n

1nj x QED

Limiting Values : x << 1

2

0 32 1 !! 2!2

s sn

ns

s

x xj x

n n s

2

10

2 1 !!

1 / 2 ! 2

s s

n ns s

n xy x

x n s

For x << 1 :

2 1 !!

n

n

xj x

n

1

2 1 !!n n

ny x

x

01n

Limiting Values : x >> n ( n + 1 ) / 2

2 1~ cos

2 2J z z

z

2 1~ sin

2 2Y z z

z

1/22n nf x F xx

1/2~2n nj x J x

x

1

cos 12

x nx

1~ sin

2n

nj x x

x

1/2~2n ny x Y x

x

1

sin 12

x nx

1

~ cos2n

ny x x

x

1n n nh x j x i y x ~ exp

2

i ni x

x

2n n nh x j x i y x ~ exp

2

i ni x

x

Travelling spherical waves

Standing spherical waves

Orthogonality & Zeros

22

1

0

1

2

a

i j i j id J J a Ja a

1/22n nf x F xx

22 2

1

0

2 21

2

an i n j n i

n n i n n j i j n ni

r rd r r j j a j

a a a

22 3

1

0

1

2

a

n n i n n j i j n ni

r rd r r j j a j

a a

Set r .

Note: n i for jn is numerically the same as n+1/2, i for Jn+1/2, .

Zeros of Spherical 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 14.7.1. Particle in a Sphere

Schrodinger eq. for free particle of mass m in a sphere of radius a :

22

2V E

m

with

0 r aV

r a

0r a

Radial eq. for r a : 2

2

120

l lR R k R

r r

2 2

2

kE

m

l lR A j kr B y kr

R is regular at r = 0 B = 0

nl l l n

rR A j

a

0r a

R l nk

a

2 2

22l n

n lEm a

0

sin xj x

x 0 n n

2 201

min 10 22E E

m a

2 2

22m a

quantized

2 1m n mnd x j x j x

n

Ex.14.7.12-3, 0m n

More Orthogonality :

General remarks :

1. Spatial confinement energy quantization.

2. Finite zero-point energy ( uncertainty principle ).

3. E is angular momentum dependent.

4. Eigenfunction belonging to same l but different n are orthogonal.

Modified Spherical Bessel Functions

Modified Spherical Bessel equation :

2

2 2 22

2 1 0d R d R

r r k r l l Rd r d r

Spherical Bessel equation :

2

2 2 22

2 1 0d R d R

r r k r l l Rd r d r

1/22n nf x F xx

1/22n ni x I x

x

1/2

2

2n nk x I xx

Caution : 1/2

2n nk x I x

x

Recurrence Relations

1/22n ni x I xx

1/2

2n nk x I x

x

1 1

2I x I x I x

x

1 1 2I x I x I x

1 1

2 1n n n

ni x i x i x

x

1 11 2 1n n nn i x n i x n i x

1 1

2 1n n n

nk x k x k x

x

1 11 2 1n n nn k x n k x n k x

i0(x), i1(x), i2(x), k0(x), k1(x), k2(x)

0

sinh xi x

x

2 2 3

sinh 3 cosh 3sinhx x xi x

x x x

1 2

1 1xk x ex x

2 2 3

1 3 3xk x ex x x

1 2

cosh sinhx xi x

x x

0

xek x

x

Mathematica

Limiting Values

For x << 1 :

2 1 !!

n

n

xi x

n

1

2 1 !!n n

nk x

x

For x >> 1 :

~2

x

n

ei x

x

~x

n

ek x

x

Example 14.7.2. Particle in a Finite Spherical Well

Schrodinger eq. for free particle of mass m in a well of radius a :

22

2V E

m

with 0 0

0

V r aV

r a

0

0

r

r

regular

Radial eq. : 2

2

120

l lR R k r R

r r

22

2

k rE V r

m

Bound states :V0 < E < 0

in lR r A j kr

Numerical solution

2 2

0 02

kE V

m

r a

out lR r B k r2 2

02

Em

r a

in outR a R a l lA j ka B k a

in outR a R a l lA k j ka B k a

Smooth connection :