1

Chapter 6 Bessel functions

Bessel functions appear in a wide variety of physical problems. For example, separation of the Helmholtz or wave equation in circular cylindrical coordinates leads to Bessel’s equation.

0][''' 222 ynxxyyx

Bessel’s eq.

The solutions of the Bessel’s eq. are called Bessel functions.In Chapter 3, we get the series solution of the above eq.

0 0

202

2

0 )2

()!(!

1)1(!2)!(!2

!)1()(j j

jnjnj

jnj x

jnjna

jnjxn

axy

0

2)2

()!(!

)1()(s

nss

nx

nssxJ

)(!20 xJna nn

2

6.1 Bessel Functions of the First Kind, )(xJv

* Generating function, integer order , J n (x)

Although Bessel functions are of interest primarily as solutions of differential equations, it is instructive and convenient to develop them from a completely different approach, that of the generating function. Let us introduce a function of two variables,

.),( )1)(2( ttxetxg (6.1)

3

Expanding it in a Laurent series, we obtain

.)()1)(2(

n

nn

ttx txJe

The coefficient of t n , J n , is defined to be Bessel function of the first kind of integer order n. Expanding the exponentials, we have

.!2

)1(!2 00

22

s

sss

r

rrtxxt

stx

rtxee

Setting n =r- s , yields

.!

)2

()1()!(20

s sn

sss

snsn

stx

sntx

Since

,0)!(

1

sn sn ( note: 0)!(

1 m for a positive integer m)

(6.2)

(6.3)

(6.4)

4

we have .000

n ss ns sn

The coefficient of t n is then

.)!1(2!2

)2

()!(!

)1()( 2

2

0

2

nx

nxx

snsxJ n

n

n

n

s

sns

n

This series form exhibits the behavior of the Bessel function J n for small x. The results for J 0 , J 1 , and J 2 are shown in Fig.6.1. The Bessel function s oscillate but are not periodic.

Figure 6.1 Bessel function, )(0 xJ )(1 xJ , and )(2 xJ

.

(6.5)

5

Eq.(6.5) actually holds for n < 0 , also giving

0

2)2

()!(!

)1()(s

nss

nx

nssxJ

Since the terms for s<n (corresponding to the negative integer (s-n) ) vanish, the series can be considered to start with s=n . Replacing s by s + n , we obtain

).()1()2

()!(!

)1()(0

2 xJxnss

xJ nn

s

nsns

n

These series expressions may be used with n replaced by v to define J v and J -v for non-integer v .

* Recurrence relations

Differentiating Eq.(6.1) partially with respect to t , we find that

,)()11(21),( 1)1)(2(

2

n

nn

ttx txnJet

xtxgt (6.9)

(6.6)

(6.7 )

6

and substituting Eq.(6.2) for the exponential and equating the coefficients of t n- 1 , we obtain

).(2)()( 11 xJxnxJxJ nnn (6.10)

This is a three-term recurrence relation. On the other hand, differentiating Eq.(6.1) partially with respect to x , we have

.)()1(21),( )1)(2(

n

nn

ttx txJet

ttxgx

Again, substituting in Eq.(6.2) and equating the coefficients of t n , we obtain the result

).(2)()( 11 xJxJxJ nnn (6.12)

(6.11)

As a special case,

).()( 10 xJxJ (6.13)

7

Adding Eqs.(6.10) and (6.12) and dividing by 2, we have

).()()(1 xJxJxnxJ nnn

Multiplying by x n and rearranging terms produces

).()( 1 xJxxJxdxd

nn

nn

Subtracting Eq.(6.12) from (6.10) and dividing by 2 yields

).()()(1 xJxJxnxJ nnn

Multiplying by x -n and rearranging terms, we obtain

).()( 1 xJxxJxdxd

nn

nn

(6.17)

(6.14)

(6.15)

(6.16)

8

).(2)()( 11 xJxnxJxJ nnn

relations recurrence basic thesatisfies )(J functions ofset a If n x

*Bessel’s differential equation

functions. Bessel are )(J i.e., equation., sBessel'satisfy indeed )(Jthen nn xx

In particular, we have shown that the functions Jn defined by our generating functions, satisfy Bessel’s eq., and thus are indeed Bessel functions

0][''' 222 ynxxyyx

).(2)()( 11 xJxJxJ nnn

Please verify the follow result in class.

9

• Integral representation A particular useful and powerful way of treating Bessel functions employs

integral representations. If we return to the generating function (Eq. (6.2)), and substitute t = e iθ ,

),3sin)(sin)((2)4cos)(2cos)((2)(

31

420sin

xJxJixJxJxJeix

in which we have used th e relations

(6.23)

,sin)(2))(()()(

1

111

xiJeexJexJexJ iiii

(6.24)

,cos)(2)()( 22

22

2 xJexJexJ ii

and so on.

10

In summation notation

,)12cos()(2)sinsin(

,)2cos()(2)()sincos(

112

120

nn

nn

nxJx

nxJxJx

equating real and imaginary parts, respectively. It might be noted that angleθ (in radius) has no dimensions. Likewise sinθ has no dimensions and function cos(xsinθ) is perfectly proper from a dimensional point of view. By employing the orthogonality properties of cousine and sine,

nmdmn

2coscos

0 (6.26a)

nmdmn

2sinsin

0 (6.26b)

(6.25)

in which n and m are positive integers (zero is excluded), we obtain

11

oddn

evennxJdnx n

0

)(cos)sincos(1

0

(6.27)

oddnxJ

evenndnx

n )(

0sin)sinsin(1

0

If these two equations are added together

,)sincos(1

sin)sinsin(cos)sincos(1)(

0

0

dxn

dnxnxxJ n

,3,2,1,0n

As a special case,

.)sincos(1)(00

dxxJ (6.30)

(6.28)

(6.29)

12

Nothing that )sincos( x repeats itself in all four quadrants ( 21 , ,

44 , ), we may write Eq. (6.30) as

.)sincos(21)(

2

00

dxxJ

On the other hand, )sinsin( x reverses its sign in the third and fourth quadrants

.0)sinsin(21 2

0

dx

Adding Eq. (6.30a) and i times Eq. (6.30b), we obtain the complex exponential

representation

.21

21)(

2

0

cos2

0

sin0

dedexJ ixix (6.30c)

(6.30a)

so that

(6.30b)

This integral representation (Eq. (6.30c)) may be obtained somewhat more directly by employing contour integration.

13

• Example 6.11 Fraunhofer Diffraction, Circular Aperture

In the theory of diffraction through a circular aperture we encounter the integral .~

0

2

0

cosa ibr rdrde

for , the amplitude of the diffracted wave. Here

plane of the circular aperture of radius a, and point on a screen below the circular aperture relative to the normal through the center point. The parameter b is given by

sin2

b

with Fig. 6.2 From Eq. (6.30c) , we get

.)(2~0 0a

rdrbrJ (6.33)

(6.31)

is an azimuth angle in the , is the angle defined by a

(6.32)

the wavelength of the incident wave. The other symbols are defined by

14Figure 6.2 Fraunhofer diffraction –circular aperture

15

Equation (6.15) enables us to integrate Eq. (6.33) immediately to obtain

).sin2(sin

~)(2~ 112

aJaabJ

bab

The intensity of the light in the diffraction pattern is proportional to 2 and

212

sinsin2~

aJ (6.35)

(6.34)

6.2 Orthogonality

0)()()()(

eq. Bessel the,n rather tha isargument theIf

2222

22

kJvkkJddkJ

dd

xk

vvv

functions Bessel ofity orthogonal theprovecan one ),/(J toJ ofargument theinto and parameters gintroducinBy

v

vm

aa

vm

16

For v > 0, J v (0)=0. Thus, for a finite interval [0, a ], when vm zero of J v (i.e. 0)( vmvJ

if m ≠ n , .0)()(0

a

vnvvmv da

Ja

J This gives us orthogonality over the interval [0, a ].

is the m th ), we are able to have

(6.49)

* Normalization The normalization result may be written as

.)(2

)( 21

2

0

2

vmv

a

vmv Jada

J

(6.50)

* Bessel series If we assume that the set of Bessel functions )( aJ vmv is complete, then any well-behaved function )(f may be expanded in a

( v fixed, m =1,2,… )

Bessel series

17

1

)()(m

vmvvm aJcf ,

,0 a

.1v

The coefficients c vm are determined by using Eq.(6.50),

.)()()(

202

12

a

vmvvmv

vm da

JfJa

c

* Continuum form If a → ∞, then the series forms may be expected to go over into integrals. The discrete roots become a continuous variable . A key relation is the Bessel function closure equation

),(1)()(0

dJJ vv

.21v (6.59)

(6.51)

(6.52)

18

Figure 6.3 Neumann functions , , )(0 xN

,

)(1 xN , and )(2 xN

19

6.3 Neumann function, Bessel function of the second kind, )(xNv

From the theory of the differential equations it is known that Bessel’s equation has two independent solutions, Indeed, for non-integral order v we have already found two solutions and labeled them and ,using )(xJv

)(xJ v

the infinite series (Eq. (6.5)). The trouble is that when v is integral Eq.(6.8) holds and we have but one independent solution. A second solution may be developed by the method of Section 3.6. This yields a perfectly good solution of Bessel’s equation but is not the usual standard form.

Definition and series form As an alternate approach, we have the particular linear combination of ' )(xJv

and )(xJ v

.sin

)()(cos)(

v

xJxJvxN vvv

(6.60)

20

This is Neumann function (Fig. 6.3). For nonintegral v ,

)(xNv

Bessel’s equation, for i t is a linear combination of known solutions, )(xJv

and )(xJ v

To verify that )(xNv

kind, actually does satisfy Bessel’s equation for integral n , we may process

clearly satisfies

, our Neumann function or Bessel function of the second

as follows. L’Hospital’s rule applied to Eq. (6.60) yields

.)()1()(1

coscos)(sin

sin)()()(cos)()(

nv

vnv

nv

vvn

nv

vvn

vxJ

vxJ

nvJvJnxJn

vdvdxJxJvdvdxN

(6.65)

21

Differentiating Bessel’f equation for with repect to v , we have

.2)()()( 222

22

vvvv vJ

vJvx

vJ

dxdx

vJ

dxdx

Multiplying the equation for )(xJ v by (-1) v , substracting from the equation

)(xJv (as suggested by Eq. (6.65)), and taking the limit nv , we obtain

.)1(2)( 222

22

nn

nnnn JJnNvxNdxdxN

dxdx

For nv , an integer, the right-hand side vanishes by Eq. (6.8) and

)(xNn

is seen to be a solution of Bessel’s equation. The most general solution for

).()()( xBNxAJxy vv (6.68)

(6.66)

for

(6.67)

any v can be written as

Example Coaxial Wave Guides We are interested in an electromagnetic wave confined between

the conducting cylindrical surfaces a and b . Most of the mathematics is concentric ,

worked out in Section 3.3. From EM knowledge,

)(xJ

22

.0222 zz EcE ZE( : electrical field along z axis)

ikzimz eePE )(

.0)()( 2222 PmkcddP

dd

This is the Bessel equation. If ,the solution is finiteP )0( )(mJ

with 2222 kc . But, for the coaxial wave guide one generalization

Let , we have

is needed. The origin is now excluded ( ). Hence the 0 ba 0

Neumann function may not be excluded.

)(mN

),,,( tzEz becomes

.)()(,

)( nm

tkziimmnmnmmnz eeNcJbE

With the condition

,0zH (6.80)

(6.79)

we have the basic equatios for a TM (transverse magnetic ) wave.

23

The (tangential) electric field must vanish at the conducting surfaces (Direchlet

0)()( aNcaJb mnmnmmn (6.81)

0)()( bNcbJb mnmnmmn these transcendental equations may be solved for )( mn and the ratio . mnmn bc

From the relation

22

22

c

k

be propagated (in this TM mode) is

2k

,c

the cutoff frequency of the wave guide.

boundary condition) or

(6.82)

(6.83)

(6.84)

and since must be positive for a real wave, the minimum frequency that will

with fixed by the boundary conditions, Eqs. (6.81) and (6.82). This is

24

6.4 Hankel function

Many authors perfer to introduce the Hankel functions by means of integral representations and then use them to define the Neumann function, .)(zNm

We here introduce them a simple way as follows. As we have already obtained the Neumann function by more elementary (and less powerful) techniques, we may use it to define the Hankel functions, )()1( xHv

and )()2( xHv :

)()()()1( xiNxJxH vvv (6.85)

)()()()2( xiNxJxH vvv (6.86)

This is exactly analogous to taking

.sincos ie i (6.87)

25

For real arguments and are complex conjugates. The extent of the analogy will be seen better when the asymptotic forms are considered . Indeed, it is their asymptotic behavior that makes the Hankel functuions useful!

)1(vH

)2(vH

6.5 Modified Bessel function , and )(xIv )(xKv The Helmholtz equation,

022 kseparated in circular cylindrical coordinates, leads to Eq. (6.22a), the Bessel

)( kJv

and )( kNv

and any linear combination such as the Hankel functions )()1( kHv

and )()2( kHv

part of wave phenomena. If instesd we have a diffusion problem, then the Helmholtz equation is replaced by

022 k . (6.88)

equation. Equation (6.22a) is satisfied by the Bessel and Neumann functions

.Now the Helmholtz equation describes the space

26

The analog to Eq. (6.22a) is

.0)()()()( 2222

22

kYvkkY

ddkY

dd

vvv

The Helmholtz equation may be transformed into the diffusion equation by the transformation . Similarly, changes Eq. (6.22a) into Eq. (6.89) and shows that

ikk

ikk

)()( ikZkY vv

The solution of Eq. (6.89) are Bessel function of imaginary argument. To obtain a solution that is regular at the origin, we take as the regular Bessel

function vZ

vJ . It is customary (and convenient) to choose the normalization so that

(Here the variable is being replaced by x for simplicity.) Often this is written as

k

).()( 22 iv

ivv xeJexI (6.91)

(6.89)

(6.90) ).()()( ixJixIkY vv

vv

27

Series form In the terms of infinite series this is equivalent to removing the s)1(

sign in Eq. (6.5) and writing

,)2

()!(!

1)(

,)2

()!(!

1)(

0

2

0

2

s

vsv

s

vsv

xvss

xI

xvss

xI

The extra vi normalization cancels the vi from each term and leaves

)(xIv real. For integral v this yields

).()( xIxI nn Recurrence relations The recurrence relations satisfied by )(xIv

expansions, but it is easier to work from the existing recurrence relations for )(xJv . Let us replace x by –ix and rewrite Eq. (6.90) as

).()( ixIixJ vv

v

(6.94)

(6.92)

(6.93)

may be developed from the series

28

Then Eq. (6.10) becomes

).(2)()( 11

11 ixIi

xvixIiixIi v

vv

vv

v

Repalcing x by ix , we have a recurrence relation for , )(xIv

).(2)()( 11 xIxvxIxI vvv

Equation (6.12) transforms to

).(2)()( 11 xIxIxI vvv

From Eq. (6.93) it is seen that we have but one independent solution when v is an integer, exactly as in the Bessel function solution of Eq. (6.108) is essentially a matter od convenience. We choose to define a second solution in terms of the Hankel function

vJ

)1(

vH by

.)()(2

)(2

)( 1)1(1 ixiNixJiixHixK vvv

vv

v (6.97)

(6.95)

(6.96)

29

The factor makes real when x is real. Using Eqs. (6.60) and (6.90), we may transform Eq. (6.97) to

1vi

)(xKv

,sin

)()(2

)(

v

xIxIxK vvv

analogous to Eq. (6.60) for The choice of Eq. (6.97) as a definition is somewhat unfortunate in that the function does not satisfy the same recurrence relations as . To avoid this annoyance other authors have included an additional factor of cos . This permits satisfy the same recurrence relations as , but it has the disadvantage of making for

)(xNv .)(xKv

)(xIvn )(xKv

)(xIv 0)( xKv

,,, 25

23

21v

To put the modified Bessel functions and in proper perspective, we introduce them here because:

)(xIv )(xKv

(6.98)

1. These functions are solutions of the frequently encountered modified Bessel equation.

2. They are needed for specific physical problems such as diffusion problems.

30

Figure 6.4 Modified Bessel functions

6.6 Asmptotic behaviors Frequently in physical problems there is a need to know how a given Bessel or modified Bessel functions for large values of argument, that is, the asymptotic behavior. Using the method of stepest descent studied in Chapter 2, we are able to derive the asymptotic behaviors of Hankel functions ( see page 450 in the text book for details) and related functions:

31

1.

2)

21(exp2)()1(

vzi

zzHv

.2arg z

2. The second kind Hankel function is just the complex conjugate of the

2)

21(exp2)()2(

vzi

zzHv

.arg2 z (6.100)

(6.99)

first (for real argument),

3. Since is the real part of )(zJv )()1( zHv

2)

21(cos2)(

vz

zzJv .arg z

4. The Neumann function is the imaginary part of )()1( zHv , or

2)

21(sin2)(

vz

zzNv (6.102)

(6.101)

5. Finally, the regular hyperbolic or modified Bessel function )(zIv

is given by

)()( izJizI vv

v

or z

ezIz

v 2)(

.2

arg2

z (6.104)

(6.1 0 3)