In the previous post we introduced the Stirling formula, which is an accurate approximation for factorials even for small numbers. In this post we will look closely at an other extension of the concept of factorials, The gamma function. But unlike the factorial, the gamma function is more broadly defined for all complex numbers other than non-positive integers. In this post we will only study the behavior of this faction over .
The gamma function is defined as follows
1- Domain of definition
The Gamma function is defined over
Indeed, we have the function is continuous and positive over .
- near 0
We can deduce then that
In other terms
2- we have
By recursion it can be shown that
3- is of over
Let’s consider the function
- We have
- is continuous
And since the function is integrable over .
4- is convex over
Therefore is convex.
5- such that
- existence of
By applying Rolle’s theorem, we find that
- uniqueness of
Since then is strictly increasing. Therefore is unique.
It follows also that:
6- near and
We can verify our results with some python code:
1 from scipy import gamma 2 3 gamma(0) 4 # inf 5 6 gamma(inf) 7 # inf
Gamma function is considered as the “correct” substitute for the factorial in various integrals, which seems to come more or less from its integral definition.