Page 781 - Handbook Of Integral Equations

Asymptotic expansions as |x| →∞
M–1

2 4x – 2νπ – π m –2m –2M
J ν (x)= cos (–1) (ν,2m)(2x) + O(|x| )
πx 4
m=0
M–1
4x – 2νπ – π m –2m–1 –2M–1

– sin (–1) (ν,2m + 1)(2x) + O(|x| ) ,
4
m=0

M–1
2 4x – 2νπ – π m –2m –2M
Y ν (x)= sin (–1) (ν,2m)(2x) + O(|x| )
πx 4
m=0
M–1
4x – 2νπ – π m –2m–1 –2M–1

+ cos (–1) (ν,2m + 1)(2x) + O(|x| ) ,
4
m=0
1
1 2 2 2 2 2 Γ( + ν + m)
2
where (ν, m)= (4ν – 1)(4ν – 3 ) ... [4ν – (2m – 1) ]= .
1
2 2m m! m! Γ( + ν – m)
2
For nonnegative integer n and large x,
√ n –2
πx J 2n (x)=(–1) (cos x + sin x)+ O(x ),
√ n+1 –2
πx J 2n+1 (x)=(–1) (cos x – sin x)+ O(x ).
Asymptotic for large ν (ν →∞).

1 ex ν 2 ex –ν
J ν (x) → √ , Y ν (x) → – ,
2πν 2ν πν 2ν
where x is fixed,
2 1/3 1 2 1/3 1
J ν (ν) → , Y ν (ν) → – .
3 2/3 Γ(2/3) ν 1/3 3 1/6 Γ(2/3) ν 1/3
Zeros of Bessel functions
Each of the functions J ν (x) and Y ν (x) has infinitely many real zeros (for real ν). All zeros are
simple, possibly except for the point x =0.
The zeros γ m of J 0 (x), i.e., the roots of the equation J 0 (γ m ) = 0, are approximately given by
γ m = 2.4 + 3.13 (m – 1) (m =1, 2, ... ),

with maximum error 0.2%.

Hankel functions (Bessel functions of the third kind)
(2)
(1)
2
H (z)= J ν (z)+ iY ν (z), H (z)= J ν (z) – iY ν (z), i = –1.
ν
ν
10.7. Modified Bessel Functions
Definitions. Basic formulas
The modified Bessel functions of the first kind, I ν (x), and the second kind, K ν (x) (also called
the Macdonald function), of order ν are solutions of the modified Bessel equation
2
2
2

x y xx + xy – (x + ν )y =0
x
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 766