[Calculus] Euler's Factorial Integral
By: Tao Steven Zheng (郑涛)
【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 .
