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.
Bessel’s eq.
x 2 y' ' xy'[ x 2  n 2 ] y  0
The solutions of the Bessel’s eq. are called Bessel functions.
In Chapter 3, we get the series solution of the above eq.

n! x n  2 j
1
x
n
y ( x)  a0  (1) 2 j
 a0 2 n! (1) j
( )n  2 j
2 j!(n  j )!
j!(n  j )! 2
j 0
j 0

j
 a0 2 n n! J n ( x)
(1) s
x
J n ( x)  
( )2sn
s 0 s!( s  n)! 2

1
6.1 Bessel Functions of the First Kind,
J v (x)
* 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,
g ( x, t )  e( x 2)(t 1 t ) .
(6.1)
2
Expanding it in a Laurent series, we obtain
e
( x 2 )( t 1 t )


n
J
(
x
)
t
 n .
(6.2)
n  
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
e xt 2  e  x 2t
s
x
t
 x t


     (1) s  
.
 2  s!
r  0  2  r! s  0

r
s
r 
(6.3)
Setting n =r- s , yields


 x
 


s 0 n   s  2 
n s
s
t n s
x
t
(1) s ( ) s .
(n  s )!
2 s!
(6.4)
Since
1


n   s ( n  s )!
  0,
( note:
1
( m )!
 0 for a positive integer m)
3

we have





             .
s 0 ns
The coefficient of t
n


J n ( x) 
s 0
s  0 n  
n   s  0
is then
(1) s
x n 2 s
xn
x n 2
( )
 n  n 2
 .
s!(n  s )! 2
2 n! 2
(n  1)!
(6.5)
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, J 0 ( x ) J 1 ( x) , and J 2 ( x)
4
Eq.(6.5) actually holds for n < 0 , also giving

J n ( x) 

s 0
(1) s
x
( ) 2 s n
s!( s  n)! 2
(6.6)
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) s  n x 2 s  n
J  n ( x)  
( )
 (1) n J n ( x).
s  0 s!( s  n)! 2

(6.7 )
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

1
1 ( x 2 )( t 1 t )
g ( x, t )  x(1  2 )e

t
2
t

n 1
nJ
(
x
)
t
, (6.9)
 n
n  
5
and substituting Eq.(6.2) for the exponential and equating the coefficients of t n- 1 ,
we obtain
J n 1 ( x)  J n 1 ( x) 
2n
J n ( x).
x
(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
1 (x
g ( x, t )  (t  )e
x
2
t
2)(t 1 t )



J n ( x)t n .
n 
Again, substituting in Eq.(6.2) and equating the coefficients of t
result
J n 1 ( x)  J n 1 ( x)  2 J n ( x).
(6.11)
n
, we obtain the
(6.12)
As a special case,
J 0 ( x)   J1 ( x).
(6.13)
6
Adding Eqs.(6.10) and (6.12) and dividing by 2, we have
J n 1 ( x) 
n
J n ( x)  J n ( x).
x
(6.14)
Multiplying by x n and rearranging terms produces


d n
x J n ( x)  x n J n 1 ( x).
dx
(6.15)
Subtracting Eq.(6.12) from (6.10) and dividing by 2 yields
n
J n 1 ( x)  J n ( x)  J n ( x).
x
(6.16)
Multiplying by x -n and rearranging terms, we obtain


d n
x J n ( x)   x  n J n 1 ( x).
dx
(6.17)
7
*Bessel’s differential equation
Please verify the follow result in class.
If a set of functions J n ( x) satisfies the basic
recurrence relations
2n
J n 1 ( x)  J n 1 ( x) 
J n ( x).
x
J n 1 ( x)  J n 1 ( x)  2 J n ( x).
then J n ( x) indeed satisfy Bessel' s equation., i.e., J n ( x) are Bessel functions.
x 2 y' ' xy'[ x 2  n 2 ] y  0
In particular, we have shown that the functions Jn defined by our generating
functions, satisfy Bessel’s eq., and thus are indeed Bessel functions
8
• 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 ,
eix sin  J 0 ( x)  2( J 2 ( x) cos 2  J 4 ( x) cos 4  )
 2i ( J1 ( x) sin   J 3 ( x) sin 3  ),
(6.23)
in which we have used th e relations
J1 ( x)ei  J 1 ( x)e  i  J1 ( x)(ei  e i )
 2iJ 1 ( x) sin  ,
(6.24)
J 2 ( x)e2i  J  2 ( x)e2i  2 J 2 ( x) cos ,
and so on.
9
In summation notation

cos( x sin  )  J 0 ( x)  2 J 2 n ( x) cos( 2n ),
n 1

sin( x sin  )  2 J 2 n 1 ( x)cos( 2n  1) ,
n 1
(6.25)
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,


0

cos n cos md 

0
sin n sin md 

2

2
 nm
 nm
(6.26a)
(6.26b)
in which n and m are positive integers (zero is excluded), we obtain
10
 J n ( x)

cos( x sin  ) cos nd  
 0
0


1


 0

sin( x sin  ) sin nd  
 0
J ( x)

 n
1


n even
n odd
(6.27)
n even
n odd
(6.28)
If these two equations are added together

cos( x sin  ) cos n  sin( x sin  ) sin n d

 0
1 
  cos( n  x sin  )d , n  0,1,2,3,
 0
J n ( x) 
1
(6.29)
As a special case,
J 0 ( x) 
1


0
cos( x sin  ) d .
(6.30)
11
Nothing that cos( x sin  ) repeats itself in all four quadrants (1   , 2    
 4     , 4   ), we may write Eq. (6.30) as
1
J 0 ( x) 
2

2
0
cos( x sin  )d .
,
(6.30a)
On the other hand, sin( x sin  ) reverses its sign in the third and fourth quadrants
so that
1
2

2
sin( x sin  )d  0.
0
(6.30b)
Adding Eq. (6.30a) and i times Eq. (6.30b), we obtain the complex exponential
representation
1
J 0 ( x) 
2

2
0
e
ix sin
1
d 
2

2
0
eix cos d .
(6.30c)
This integral representation (Eq. (6.30c)) may be obtained somewhat more
directly by employing contour integration.
12
• Example 6.11 Fraunhofer Diffraction, Circular Aperture
In the theory of diffraction through a circular aperture we encounter the integral
~
a
0

2
eibr cos d rdr.
0
(6.31)
for  , the amplitude of the diffracted wave. Here is an azimuth angle in the
plane of the circular aperture of radius a, and  , is the angle defined by a
point on a screen below the circular aperture relative to the normal through
the center point. The parameter b is given by
b
2

sin 
(6.32)
with  the wavelength of the incident wave. The other symbols are defined by
Fig. 6.2 From Eq. (6.30c) , we get
a
 ~ 2  J 0 (br )rdr.
0
(6.33)
13
Figure 6.2 Fraunhofer diffraction –circular aperture
14
Equation (6.15) enables us to integrate Eq. (6.33) immediately to obtain
2ab
a
2a
~
J1 ( ab) ~
J1 (
sin  ).
2
b
sin 

(6.34)
2
The intensity of the light in the diffraction pattern is proportional to  and
 J 2a  sin  
2 ~  1

sin



2
(6.35)
6.2 Orthogonality
If the argument is k rather tha n x, the Bessel eq.
d2
d
2 2
2

J
(
k

)


J
(
k

)

(
k


v
) J v (k )  0
v
v
2
d
d
2
By introducin g parameters a and m into the argument of J v to
J v ( vm  / a), one can prove the orthogonal ity of Bessel functions
15
For v > 0, J v (0)=0. Thus, for a finite interval [0, a ], when  vm is the m th
zero of J v (i.e. J v ( vm )  0 ), we are able to have
if m ≠ n ,

a
0
J v ( vm

a
) J v ( vn

a
) d  0.
(6.49)
This gives us orthogonality over the interval [0, a ].
* Normalization
The normalization result may be written as

a
0
 
a2

2


J
(

)

d


J
(

)
.
v 1
vm
 v vm a 
2
2
(6.50)
* Bessel series
If we assume that the set of Bessel functions J v ( vm  a) ( v fixed, m =1,2,… )
is complete, then any well-behaved function f (  ) may be expanded in a
Bessel series
16

f (  )   cvm J v ( vm
m 1

a
) ,
0    a,
v  1.
(6.51)
The coefficients c vm are determined by using Eq.(6.50),
cvm
2
 2
2
a J v 1 ( vm )

a
0
f (  ) J v ( vm

a
) d .
(6.52)
* 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


0
J v () J v (  ) d 
1
 (   ), v   1 .

2
(6.59)
17
Figure 6.3 Neumann functions , N 0 ( x) , N1 ( x) , and N 2 ( x)
18
6.3 Neumann function, Bessel function of the second kind, N v (x)
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 themJ v (x) and J  v (x) ,using
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 J'v (x)
and J  v (x)
cos vJ v ( x)  J  v ( x)
N v ( x) 
.
sin v
(6.60)
19
This is Neumann function (Fig. 6.3). For nonintegral v , N v (x) clearly satisfies
Bessel’s equation, for i t is a linear combination of known solutions, J v (x)
and J  v (x)
To verify that N v (x) , our Neumann function or Bessel function of the second
kind, actually does satisfy Bessel’s equation for integral n , we may process
as follows. L’Hospital’s rule applied to Eq. (6.60) yields
N n ( x) 
(d dv)cos vJ v ( x)  J  v ( x)
(d dv) sin v
vn
  sin nJ n ( x)  cos n J v v  J  v v 

 cos n
vn
1  J v ( x)
n J  v ( x ) 
 
 (1)
.

  v
v  v  n
(6.65)
20
Differentiating Bessel’f equation for J  (x) with repect to v , we have
d 2 J  v
d J  v
2
2 J  v
x
(
)

x
(
)

(
x

v
)
 2vJ v .
(6.66)
2
dx
v
dx v
v
Multiplying the equation for J  v (x) by (-1) v , substracting from the equation
2
for J v (x) (as suggested by Eq. (6.65)), and taking the limitv  n , we obtain


d2
d
2n
2
2
x 2 Nn  x Nn  ( x  v ) Nn 
J n  (1)n J  n .
dx
dx

2
(6.67)
For v  n , an integer, the right-hand side vanishes by Eq. (6.8) and N n (x)
is seen to be a solution of Bessel’s equation. The most general solution for
any v can be written as
y( x)  AJ v ( x)  BN v ( x).
(6.68)
Example Coaxial Wave Guides
We are interested in an electromagnetic wave confined between concentric ,
the conducting cylindrical surfaces   a and   b . Most of the mathematics is
worked out in Section 3.3. From EM knowledge,
21
 2 E z   2 c 2 E z  0.
( EZ: electrical field along z axis)
Let Ez  P(  )eim eikz , we have



d
dP
(
)  ( 2 c 2  k 2 )  m 2 P  0.
d
d
This is the Bessel equation. If P(   0)  finite ,the solution is J m ( )
with  2   2 c 2  k.2 But, for the coaxial wave guide one generalization
is needed. The origin   0 is now excluded ( 0  a    b ). Hence the
Neumann function N m ( ) may not be excluded. Ez (  , , z, t ) becomes
Ez   bmn J m ( )  cmn N mn ( )e  im ei ( kz t ) . (6.79)
m, n
With the condition
H z  0,
we have the basic equatios for a TM (transverse magnetic ) wave.
(6.80)
22
The (tangential) electric field must vanish at the conducting surfaces (Direchlet
boundary condition) or
bmn J m (a)  cmn N mn (a)  0
(6.81)
bmn J m (b)  cmn N mn (b)  0
(6.82)
these transcendental equations may be solved for  ( mn ) and the ratio cmn bmn .
From the relation
2

k2  2  2
c
(6.83)
and since k 2 must be positive for a real wave, the minimum frequency that will
be propagated (in this TM mode) is
  c ,
(6.84)
with  fixed by the boundary conditions, Eqs. (6.81) and (6.82). This is
the cutoff frequency of the wave guide.
23
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, N m (z ) .
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, H v(1) ( x )
and H v( 2 ) ( x) :
H v(1) ( x)  J v ( x)  iN v ( x)
(6.85)
H v( 2 ) ( x)  J v ( x)  iN v ( x)
(6.86)
This is exactly analogous to taking
e  i  cos  i sin  .
(6.87)
24
( 2)
For real argumentsH v(1) and H v 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!
6.5 Modified Bessel function , I v (x) and K v (x)
The Helmholtz equation,
 2  k 2  0
separated in circular cylindrical coordinates, leads to Eq. (6.22a), the Bessel
equation. Equation (6.22a) is satisfied by the Bessel and Neumann functions
Nv (k ) and J v (k ) and any linear combination such as the Hankel functions
H v(1) (k ) and H v( 2) (k ) .Now the Helmholtz equation describes the space
part of wave phenomena. If instesd we have a diffusion problem, then the
Helmholtz equation is replaced by
2  k 2  0
.
(6.88)
25
The analog to Eq. (6.22a) is
d2
d
2 2
2

Y
(
k

)


Y
(
k

)

(
k


v
)Yv (k )  0.
v
2 v
d
d
2
(6.89)
The Helmholtz equation may be transformed into the diffusion equation by the
transformation k  ik . Similarly, k  ik changes Eq. (6.22a) into Eq. (6.89)
and shows that
Yv (k )  Z v (ik )
The solution of Eq. (6.89) are Bessel function of imaginary argument. To obtain
a solution that is regular at the origin, we take Z v as the regular Bessel function
J v . It is customary (and convenient) to choose the normalization so that
Yv (k )  I v ( x)  i  v J v (ix ).
(6.90)
(Here the variable k is being replaced by x for simplicity.) Often this
is written as
I v ( x)  e  iv 2 J v ( xei 2 ).
(6.91)
26
Series form
In the terms of infinite series this is equivalent to removing the(1) s
sign in Eq. (6.5) and writing

1
x 2s v
I v ( x)  
( ) ,
s  0 s!( s  v )! 2

(6.92)
1
x 2s v
I  v ( x)  
( ) ,
s  0 s!( s  v )! 2
v
The extra i  v normalization cancels the i from each term and leaves
I v (x) real. For integral v this yields
I n ( x)  I  n ( x).
(6.93)
Recurrence relations
The recurrence relations satisfied by I v (x) may be developed from the series
expansions, but it is easier to work from the existing recurrence relations for
J v (x) . Let us replace x by –ix and rewrite Eq. (6.90) as
J v ( x)  i v I v (ix ).
(6.94)
27
Then Eq. (6.10) becomes
i
v 1
I v 1 (ix )  i
v 1
2v v
I v 1 (ix ) 
i I v (ix ).
x
Repalcing x by ix , we have a recurrence relation for I v (x),
I v 1 ( x)  I v 1 ( x) 
2v
I v ( x).
x
(6.95)
Equation (6.12) transforms to
I v 1 ( x)  I v 1 ( x)  2I v ( x).
(6.96)
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 J v solution of Eq. (6.108) is
essentially a matter od convenience.
We choose to define a second solution in terms of the Hankel function H v(1) by
K v ( x) 

2
i
v 1
H (ix ) 
(1)
v

2
i v 1 J v (ix )  iN v (ix ).
(6.97)
28
The factor i v 1 makes K v (x) real when x is real. Using Eqs. (6.60) and (6.90),
we may transform Eq. (6.97) to
K v ( x) 
 I  v ( x)  I v ( x)
,
2
sin v
(6.98)
analogous to Eq. (6.60) for N v (x) The. choice of Eq. (6.97) as a definition is
somewhat unfortunate in that the function K v (x) does not satisfy the same
recurrence relations as I v (x) . To avoid this annoyance other authors have
included an additional factor of cos n . This permits K v (x) satisfy the same
recurrence relations as I v (x) , but it has the disadvantage of making K v ( x)  0
for v  12 , 32 , 52 ,
To put the modified Bessel functions I v (x) and K v (x) in proper perspective,
we introduce them here because:
1. These functions are solutions of the frequently encountered
modified Bessel equation.
2. They are needed for specific physical problems such as diffusion
problems.
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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
30
text book for details) and related functions:
1.
2.

2
1  
exp i  z  (v  )      arg z  2 .
(6.99)
z
2
2


The second kind Hankel function is just the complex conjugate of the
first (for real argument),
H v(1) ( z ) 
 
2
1  
exp  i  z  (v  )    2  arg z   .
z
2 2 
 
3. Since J v (z ) is the real part of H v(1) ( z )
H v( 2 ) ( z ) 
J v ( z) 
2
1 

cos  z  (v  ) 
z
2 2

(6.100)
   arg z   .
(6.101)
(1)
4. The Neumann function is the imaginary part of H v ( z ) , or
2
1 

sin  z  (v  ) 
z
2 2

5. Finally, the regular hyperbolic or modified Bessel function I v (z )
Nv ( z) 
is given by
or
I v ( z )  i  v J v (iz )
Iv ( z) 
ez
2z
(6.102)
(6.1 0 3)


2
 arg z 

2
.
(6.104)
31