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Behaviors of Bessel functions for large arguments:
(
)
2
π vπ
J v (x) ≈
cos x − −
πx
4
2
(
)
2
π vπ
Y v (x) ≈
sin x − −
πx
4
2
e x
4v 2 − 1
I v (x) ≈ √
1 −
2πx
8x
π −x
4v 2 − 1
K v (x) ≈
e
1 +
2x
8x
300
Appendix A
Properties of Bessel functions:
d
d
[J 0 (mx)] = −mJ 1 (mx),
[Y 0 (mx)] = −mY 1 (mx)
dx
dx

d
d

[I 0 (mx)] = mI 1 (mx),
[K 0 (mx)] = −mK 1 (mx)
dx
dx
A.3.2 Bessel Functions of the First Kind [1]
x
J 0 (x)
J 1 (x)
0.0
1.0000
0.0000
0.1
0.9975
0.0499
0.2
0.9900
0.0995
0.3
0.9776
0.1483
0.4
0.9604
0.1960
0.5
0.9385
0.2423
0.6
0.9120
0.2867
0.7
0.8812
0.3290
0.8
0.8463
0.3688
0.9
0.8075
0.4059
1.0
0.7652
0.4400
1.1
0.7196
0.4709
1.2
0.6711
0.4983
1.3
0.6201
0.5220
1.4
0.5669
0.5419
1.5
0.5118
0.5579
1.6
0.4554
0.5699
1.7
0.3980
0.5778