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534 CHAPTER 15 Special Functions and Eigenfunction Expansions

THEOREM 15.8 Factorial Property of the Gamma Function

If x > 0 then

(x + 1) = x
(x).

−t
x
Proof This is a straightforward integration by parts, with u = t and dv = e dt:
∞

x −t

(x + 1) = t e dt
0

∞
x −t ∞ x−1 −t
=[t (−e )] − xt (−1)e dt
0
0

∞
e dt = x
(x).
= x t x−1 −t
0
To see why this is called the factorial property, first compute
∞

−t

(1) = e dt = 1.
0
By Theorem 15.8,

(2) = 1
(1) = 1,
(3) = 2
(2) = 2!,

(4) = 3
(3) = 3 · 2 = 3!,
(5) = 4
(4) = 4 · 3!= 4!,
and, for any positive integer n,
(n + 1) = n!.
The gamma function can be extended to negative noninteger values by rewriting the factorial
property as
1

(x) =
(x + 1)
x
for x >0. If −1< x <0, then x +1>0, so
(x +1) is defined and we can
(x)=(1/x)
(x +1).
Once we have defined
(x) for −1 < x < 0, then we can use this strategy again to
(x) for
−2< x <−1. In this way we can walk to the left over the real line, defining
(x) for all negative
numbers except integers.
For example,
1
1

(−1/2) =
− + 1 =−2
(1/2)
−1/2 2
and
1
−3 2 4

(−3/2) =
+ 1 =−
(−1/2) =
(1/2).
−3/2 2 3 3

15.3.2 Bessel Functions of the First Kind

The differential equation
2
2
2
x y + xy + (x − ν )y = 0 (15.7)

in which ν ≥ 0, is called Bessel’s equation of order ν. This differential equation is second-
order, and the phrase “order ν” refers to the parameter ν appearing in it.

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October 14, 2010 15:20 THM/NEIL Page-534 27410_15_ch15_p505-562