Page 782 - Handbook Of Integral Equations

and are defined by the formulas

∞ 2k+ν
(x/2) π I –ν – I ν
I ν (x)= , K ν (x)= ,
k! Γ(ν + k +1) 2 sin πν
k=0
(see below for K ν (x) with ν =0, 1, 2, ... ).
The modified Bessel functions possess the properties

n
K –ν (x)= K ν (x); I –n (x)=(–1) I n (x), n =0, 1, 2, ...
2νI ν (x)= x[I ν–1 (x) – I ν+1 (x)], 2νK ν (x)= –x[K ν–1 (x) – K ν+1 (x)],
d 1 d 1
I ν (x)= [I ν–1 (x)+ I ν+1 (x)], K ν (x)= – [K ν–1 (x)+ K ν+1 (x)].
dx 2 dx 2
1
Modified Bessel functions for ν = ±n ± , where n = 0,1,2, ...
2

2 2
I 1/2 (x)= sinh x, I –1/2 (x)= cosh x,
πx πx

2 1 2 1
I 3/2 (x)= – sinh x + cosh x , I –3/2 (x)= – cosh x + sinh x ,
πx x πx x
n n
k
1 x (–1) (n + k)! n –x (n + k)!
I n+1/2 (x)= √ e k – (–1) e k ,
2πx k!(n – k)! (2x) k!(n – k)! (2x)
k=0 k=0
n k n
1 x (–1) (n + k)! n –x (n + k)!

I –n–1/2 (x)= √ e k +(–1) e k ,
2πx k!(n – k)! (2x) k!(n – k)! (2x)
k=0 k=0

π –x π 1 –x
K ±1/2 (x)= e , K ±3/2 (x)= 1+ e ,
2x 2x x
n

π –x (n + k)!
K n+1/2 (x)= K –n–1/2 (x)= e .
2x k!(n – k)! (2x) k
k=0
Modified Bessel functions ν = n, where n = 0,1,2, ...
If ν = n is a nonnegative integer, then
n–1
x
x 1 m 2m–n (n – m – 1)!
n+1
K n (x)=(–1) I n (x)ln + (–1)
2 2 2 m!
m=0
1 n x n+2m ψ(n + m +1)+ ψ(m +1)
∞
+ (–1) ; n =0, 1, 2, ... ,
2 2 m!(n + m)!
m=0
where ψ(z) is the logarithmic derivative of the gamma function; for n = 0, the first sum is dropped.
Wronskians and similar formulas.
2 1
W(I ν , I –ν )= – sin(πν), W(I ν , K ν )= – ,
πx x
2 sin(πν) 1
I ν (x)I –ν+1 (x) – I –ν (x)I ν–1 (x)= – , I ν (x)K ν+1 (x)+ I ν+1 (x)K ν (x)= ,
πx x
where W(f, g)= fg – f g.

x
x
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 767