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493
A P P E N D I X B
Bessel Functions
B.1 BESSEL FUNCTIONS OF THE FIRST KIND
Bessel functions of the first kind of integer order
are defined as the solution of theintegral equation
(B.1)
where
j
is the square root of
1. The special case reduces to
(B.2)
For a real argument
z
, the Bessel functions are real valued, continuously
differentia-ble, and bounded in magnitude by unity. The
even-numbered Bessel functions aresymmetric and the odd-numbered
Bessel functions are antisymmetric.
The Bessel function may also be expressed as the infinite
series
(B.3)
where is the
gamma function
; for integer values, .We plot
J
0
(
z
) and
J
1
(
z
) for real-valued
z
in Fig. B.1. The values of these functionsfor a subset of
z
are given in Table B.1.
Problem B.1
Using the first line of Eq. (B.1), derive the second line of the
equation.
J z( ) 1--- z sin( )cos d0=
j
------- e
jz cos( )cos d
0
= 0=
J0 z( ) 1--- z sin( )cos d0=
J z( )
J z( ) 12--- z
14--- z
2
k
k! k 1+ +( )-----------------------------------k 0=
=
k( ) k 1+( ) k!=
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494
Appendix B Bessel Functions
FIGURE B.1
Plots of Bessel functions of the first kind,
J
0
(
x
) and
J
1
(
x
).
J0(x)
J1(x)
10 68 4 2 84 60 2 10
1.0
0.5
0.0
0.5
1.0
1.5
Bes
sel f
unct
ions
of f
irst
kin
d
x
TABLE B.1
Values of Bessel Functions and Modified Bessel Functions of the
First Kind.
x J
0
(
x
)
J
1
(
x
)
I
0
(
x
)
I
1
(
x
)
0.00 1.0000 0.0000 1.0000 0.0000
0.20 0.9900 0.0995 1.0100 0.1005
0.40 0.9604 0.1960 1.0404 0.2040
0.60 0.9120 0.2867 1.0920 0.3137
0.80 0.8463 0.3688 1.1665 0.4329
1.00 0.7652 0.4401 1.2661 0.5652
1.20 0.6711 0.4983 1.3937 0.7147
1.40 0.5669 0.5419 1.5534 0.8861
1.60 0.4554 0.5699 1.7500 1.0848
1.80 0.3400 0.5815 1.9896 1.3172
2.00 0.2239 0.5767 1.1796 1.5906
2.20 0.1104 0.5560 2.6291 1.9141
2.40 0.0025 0.5202 3.0493 2.2981
2.60
0.0968 0.4708 3.5533 2.7554
2.80
0.1850 0.4097 4.1573 3.3011
3.00
0.2601 0.3391 4.8808 3.9534
3.20
0.3202 0.2613 5.7472 4.7343
3.40
0.3643 0.1792 6.7848 5.6701
3.60
0.3918 0.0955 8.0277 6.7927
3.80
0.4026 0.0128 9.5169 8.1404
4.00
0.3971
0.0660 11.3019 9.7595
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Section B.2 Modified Bessel Functions of the First Kind
495
B.2 MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND
Modified Bessel functions of the first kind of integer order
are defined as the solutionof the integral equation
(B.4)
For the special case , Eq. (B.4) reduces to
(B.5)
For a real argument
z
, the modified Bessel functions are real valued, continuously
dif-ferentiable, and grow exponentially as |
z
| increases. The even-numbered modifiedBessel functions are
symmetric and the odd-numbered ones are antisymmetric.
The modified Bessel function may also be expressed as the
infinite series
(B.6)
We plot
I
0
(
z
) and
I
1
(
z
) for real-valued
z
in Fig. B.2. The values of these functions for asubset of
z
are given in Table B.1.
I z( ) 1--- ez cos
( )cos d0
= 0=
I0 z( ) 1--- ez cos d
0
=
I z( ) 12--- z
14--- z
2 k
k! k 1+ +( )-----------------------------------k 0=
=
FIGURE B.2 Plots of modified Bessel functions of the first kind,
I0(z) and I1(z).
4 23 1 0 31 2 4
10
6
2
2
6
10
8
4
0
4
8
Mod
ifie
d B
esse
l Fun
ctio
ns o
f fir
st k
ind
x
I1(x)
I0(x)
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