3. Neumann Functions,

Bessel Functions of the 2nd Kind

Neumann Functions : cos

sin

J x J xY x

2

0 1 ! 2

s s

s

xJ x

s s

x << 1 1 1

sin 1 2

xY x

1sin

2

xY x

cos sin

limcosn

d J x d J xJ x

d d

limn n

Y x Y x

1 n

n

d J x d J x

d d

2

0 1 ! 2

s s

s

xJ x

s s

1 n

n

n

d J x d J xY x

d d

2

0

1ln

2 1 ! 2

s s

s

d J x sx xJ x

d s s

ln2

2

x

d x d e

d d

ln2 2

x x

11 1

d ss s

d

21

0

2

0

1 !2 1ln

2 ! 2

11 1

! ! 2

k nn

n nk

k k n

k

n kx xY x J x

k

xk n k

n k k

Ex.14.3.8

1 !

2

n

n

n xY x

x << 1 2

xY x

agrees with

21

0

2

0

1 !2 1ln

2 ! 2

11 1

! ! 2

k nn

n nk

k k n

k

n kx xY x J x

k

xk n k

n k k

2

0 00

2 2ln

2 ! ! 2

k k

kk

x xY x J x H

k k

1 nn H

1

1n

nj

Hj

2

00

2 2ln

2 ! ! 2

k k

kk

x xJ x H

k k

Mathematica

For x , periodic with amp x 1/2

/2 phase difference with Jn

Integral Representation

1

21 2

2cos2

11

2

xxt

Y x d tt

0

0

2cos coshY x d t x t

1Re , 0

2x

0x

1/22

1

cos2

1

xtd t

t

Ex.14.3.7Ex.14.4.8

Recurrence Relations cos

sin

J x J xY x

1 1

2J x J x J x

x

1 1 1 11 1

cos

sin

J x J x x J x J xY x Y x

x

1 1

2J x J x J x

x

1 11

cos 1

sin 1

J x J xY x

1 11

cos 1

sin 1

J x J xY x

1 1cos

sin

J x J x

1 1cos

sin

J x J

cos2

sin

J x x J x

x x

1 1

2Y x Y x Y x

x

cos

sin

J x J xY x

1 1 1 11 1

cos

sin

J x J x x J x J xY x Y x

x

1 11

cos

sin

J x J xY x

1 1

1

cos

sin

J x JY x

cos2

sin

J x x J x

x

1 1

2Y x Y x Y x

x

1 1 2J x J x J x

1 1 2J x J x J x cos

sin

J x J xY x

Since Y satisfy the same RRs for J , they are also the solutions to the Bessel eq.

Caution: Since RR relates solutions to different ODEs (of different ), it depends on their normalizations.

Wronskian Formulas

For an ODE in self-adjoint form 0p x y r x y

the Wronskian of any two solutions satisfies

,

AW u v

p x ,

u vW u v

u v

u v u v

Ex.7.6.1

Bessel eq. in self-adjoint form : 2

0x y x yx

For a noninteger , the two independent solutions J & J satisfy

AJ J J J

x

AJ J J J

x

A can be determined at any point, such as x = 0.

2

0 1 ! 2

s s

s

xJ x

s s

1

1 2

xJ x

1

2 1 2

xJ x

1

1 2

xJ x

1

2 1 2

xJ x

1

22 1 1 2

xJ J J J

2 sin

x

1sin

2 sinJ J J J

x

2 sin

A

0nA

More 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

Combining the Wronskian with the previous recurrence relations,

one gets many more recurrence relations

Uses of Neumann Functions

1. Complete the general solutions.

2. Applicable to any region excluding the origin ( e.g.,

coaxial cable, quantum scattering ).

3. Build up the Hankel functions ( for propagating waves ).

Example 14.3.1. Coaxial Wave Guides

EM waves in region between 2 concentric cylindrical

conductors of radii a & b. ( c.f., Eg.14.1.2 & Ex.14.1.26 )

For TM mode in cylindrical cavity (eg.14.1.2) :

cosi mz m m j

pE J e z

a h

2 2 0zk E

2 222 m j m j pp

ka h c

For TM mode in coaxial cable of radii a & b :

i m i l z i tz mn m mn mn m mnE c J d Y e e e

0

0

mn m mn mn m mn

mn m mn mn m mn

c J a d Y a

c J b d Y b

22 2 2

mnk lc

with cutoff mn c

Note: No cut-off for TEM modes.

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