Euler's Integral
There are two types of Euler's Integral :
Euler's integral of first kind. It is the also known as Beta Function and is defined as
for all such that
For some positive integers we can define the beta function as
Euler's integral of second kind. It is the also known as Gamma Function and is defined as
for all such that
For some positive integer , we can define the gamma function as
Contents
Relationship
Both of the Euler's integral can be related to each other by the following formula
Recall the definition of gamma function that :
Now one can write :
We can rewrite it as a double integral :
Apply the substitution we have :
Using definition of gamma and beta function we have :
Hence proved
See Also
For further details you can read the following wikis