(continued)
x
sinh x
cosh x
tanh x
2.80
2.90
8.1919
9.0596
8.2527
9.1146
0.99263
0.99396
3.00
3.50
4.00
4.50
5.00
10.018
16.543
27.290
45.003
74.203
10.068
16.573
27.308
45.014
74.210
0.99505
0.99818
0.99933
0.99975
0.99991
6.00
7.00
8.00
9.00
10.00
201.71
548.32
1490.5
4051.5
11013
201.72
548.32
1490.5
4051.5
11013
0.99999
1.00000
1.00000
1.00000
1.00000
�
�
�
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(x/2) 2
(x/2) 4
J 0 (x) = 1 −
+
− · · ·
(1!) 2
(2!) 2
x (x/2) 3
(x/2) 5
J 1 (x) = −
+
− · · ·
2
1!2!
2!3!
. . .
(x/2) v
(x/2) 2
(x/2) 4
J v (x) =
1 −
+
− · · ·
Γ(v + 1)
1!(v + 1) 2!(v + 1)(v + 2)
(x/2) 2
(x/2) 4
I 0 (x) = 1 +
+
+ · · ·
(1!) 2
(2!) 2
x (x/2) 3
(x/2) 5
I 1 (x) = −
+
− · · ·
2
1!2!
2!3!
. . .
(x/2) v
(x/2) 2
(x/2) 4
I v (x) =
1 +
+
+ · · ·
Γ(v + 1)
1!(v + 1) 2!(v + 1)(v + 2)
Appendix A
299
A.3 Bessel Functions
A.3.1 Bessel Functions and Properties [2]
Behaviors of Bessel functions for small arguments: