Can the Gamma Function Simplify Complex Integrals?

[Greg Bernhardt]

Definition/Summary

The gamma function denoted by $\Gamma (n)$ is defined by

$$\Gamma (n) = \int_{0}^{\infty} x^{n-1} e^{-x} dx$$

is convergent for real and complex argument except for 0, -1, -2, ...-k

Equations

Useful identities:
$$\Gamma(n+1)=n!$$

$$\Gamma (x+1) = x\Gamma(x)$$

$$\Gamma \left(\frac12\right) = \sqrt\pi$$

$$\Gamma(x) \Gamma(1-x) = \frac{\pi}{\sin(x\pi)}$$

Extended explanation

The gamma function comes up often in math and physics when dealing with complicated integrals. It has interesting properties, and one of them caught the eye of a Swiss mathematician Leonhard Euler. He noted that the integral in question is related to a factorial:

If n is an integer, then

$$n! = \Gamma (n+1)$$

But gamma doesn't have to be restricted to only integers, therefore the factorial of real and complex numbers is naturally extended. Another useful property of the gamma is the recurrence formula

$$\Gamma (x+1) = x\Gamma(x)$$

which allows one to obtain other values of the integral by knowing its previous values.

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[fresh_42]

The gamma function is an important function in number theory and it looks like it. But it can be surprisingly useful in integration. Have a look at this fancy integration trick:
https://www.physicsforums.com/threads/micromass-big-integral-challenge.867904/page-2#post-5451293