ABSTRACT
The gamma function, denoted Γ(x), is defined by
Γ( ) ,x e t dt xt x= >− − ∞∫ 1 0
• Properties
Γ Γ
Γ
Γ Γ
Γ Γ
( ) ( ), ( ) ( ) ( ) ! ( , , , ) ( ) ( )
x x x x
n n n n n x x
+ = >
=
+ = = = …
−
1 0 1 1
1 1 2 3 1 =
⎛
⎝
⎜
⎞
⎠
⎟
=
+ ⎛
⎝
⎜
⎞
⎠
⎟
= −
pi pi
pi
pi
/sin x
x x xx
Γ
Γ Γ Γ
1 2
2 12 2 2 1 ( ) ( )
The Laplace transform of the function f(t), denoted by F(s) or L{f(t)}, is defined
F s f t e dtst( ) ( )= − ∞∫ 0
provided that the integration may be validly performed. A sufficient condition for the existence of F(s) is that f(t) be of exponential order as t → ∞ and that it is sectionally continuous over every finite interval in the range t ≥ 0. The Laplace transform of g(t) is denoted by L g t{ ( )} or G(s).
