The gamma function is defined for all complex z except at z=0,-1,-2,-3,…, by

We see first that . Next, we can apply integration by parts to for real *x* to get:

.

Thus for an integer n, we see

Thus the gamma function is an extension of the factorial to non-integer and non-real values.

We can also compute :

.

Making the u-substitution , we get:

.

And so

and, thus for integer n:

We also have

and so on.

If we take the original integral and make the substitution , we get:

And thus for an integer n>0,

Also, according to the Bohr-Mollerup theorem, the gamma function is the only function on x>0 for which the following three properties hold:

I.

II.

III. is convex.

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Tags: Factorial, Gamma Function, Math, Monday Math

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