[Calculus] Euler's Factorial Integral
By: Tao Steven Zheng (鄭濤)
【Problem】
In 1729, Leonhard Euler (1707 - 1783) discovered the integral:
Show that the integral? is equivalent to the Gamma function
; hence, it is equivalent to
for non-negative integers
.
【Solution】
Let , then
.? Consequently
, which means
.?
When , we have?
. When
, we have?
. Thus, the original integral transforms to
or
.
This integral is called he Gamma function, and it is defined as
for non-negative integers .
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