Gamma Function Notes
The gamma function is widely used in applied mathematics and probability.
An
discussion of this function provides a review of calculus and an introduction to numerical
methods.
The gamma function, denoted by
()
x
Γ
, is defined for
1
x
>
and is related to
its argument in the following manner.
1
0
xt
e
d
t
∞
−−
Γ
=
∫
(
1
)
There is no analytical expression for arbitrary values of the argument.
The function is
frequently written in the following manner.
0
1,
0
bt
e
d
t
b
∞
−
Γ
+=
>
∫
(
2
)
There is a special case for which one can derive an analytical expression for
1
b
Γ
+
,
namely when
b
is a nonnegative integer.
Let us consider the following special cases.
Case 1:
b
= 0
The value of the gamma function is as follows
0
0
1
lim
lim 1
1
t
x
t
x
x
x
ed
t
t
e
∞
−
−
→∞
−
Γ
=
=
=
−
=
∫
∫
Case 2:
b
= 1
The value of the gamma function requires that we use the integration by parts method of
calculus.
0
0
0
2
lim
,
lim
1
t
x
tt
t
x
x
x
te dt
te
dt
u
t
dt
du dv
e dt
v
e
e dt
∞
−
Γ
=
==
⇒
⇒
=
−
⎡⎤
=
−
+
⎢⎥
⎣⎦
=
∫
∫
∫
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 Spring '11
 Neveskyi
 Derivative, Abramowitz and Stegun, Gamma function, −t dt

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