Handbook of Exact Solutions for Ordinary Differential Equations - Supplement 2 - Some Special Functions.pdf

文档介绍：Supplement 2
Some Special Functions
In Supplement 2, n is a positive integer, unless otherwise specified.
. Gamma-function
The gamma-function Γ(z) is an analitic function of z everywhere, except for the points
z =0,−1, −2, ...
For Re z>0,

∞
Γ(z)= tz−1e−t dt.
0
For −(n +1)< Re z<−n, where n is an integer,


∞n
(−1)m
Γ(z)= e−t − tz−1 dt.
m!
0 m=0
The gamma-function possesses the following properties:
Γ(z +1)=zΓ(z), Γ(n +1)=n!, Γ(1) = Γ(2) = 1.
   
ππ 1 1 π
Γ(z)Γ(−z)=−, Γ(z)Γ(1 − z)= , Γ+ z Γ− z = ,
z sin(πz) sin(πz) 2 2 cos(πz)
     
22z−1 1 33z−1/2 1 2
Γ(2z)= √Γ(z)Γ z + , Γ(3z)= Γ(z)Γ z + Γ z + ,
π 2 2π 3 3
n−1  
k
Γ(nz)=(2π)(1−n)/2nnz−1/2 Γ z + ,
n
k=0
    √
1 √ 1 π
Γ= π, Γ n + = (2n − 1)!!,
2 2n
 2    √
1 √ 1 2n π
Γ−= −2 π, Γ n −=(−1)n ,
2 2 (2n − 1)!!
Γ(z + n) Γ(z +1)
=(z) , = Cw.
Γ(z) n Γ(w + 1)Γ(z − w +1) z
c 1995 by CRC Press, Inc.
. Bessel functions Jν and Yν
. Basic Formulae
The Bessel functions of the first and second kind Jν and Yν(function Yν is also called
the Neumann function) are solutions of the Bessel equation and are defined by
the formulae
∞ k ν+2k
(−1) (x/2) Jν(x) cos πν− J−ν(x)
Jν(x)= ,Yν(x)= .
k!Γ(ν+ k +1) sin πν
k=0
The formulae for Yν(x) is valid for ν=0, ±1, ±2, ...; (see below for the integral represen-
tatio