# Derivation of Generalization of the Gamma Function

The following definite integral problem was posted on stackexchange today.

(1)

The version of the gamma function, a useful function for solving improper integrals, that I learned as an undergraduate physics major is

(2)

At first glance, it appears that the gamma function (equation (2)) can be used to solve equation (1), since a constant times a definite integral is equal to the constant times the definite integral. However, since , equation (2) can’t be used to solve equation (1); the integrands are not of the same form. But it is possible to derive a formula for solving improper integrals of the same form as equation (2) using u-substitution and the gamma function.

Equation (1) has the form

(3)

(with c and a being constants being any constant).

Let

(4)

It follows from equation (4) that

(5)

and

(6)

Applying equations (4), (5), and (6) to equation (3):

(7)

(8)

In equation (7), is just the gamma function, and is thus equal to .

Note: the bounds of integration have not changed, since when x goes to zero, from equation (4), u goes to zero as well. And when x goes to infinity, u goes to infinity for any non-zero constant a.

Finally, from equations(7) and (2) (the gamma function):

(9)

From (1), and . So the solution to (1) is:

Without using equation (8), the integral could still be solved, but it would require using integration by parts multiple times (unless there is another method that I am unaware of), which is an unnecessarily long and arduous way to evaluate the integral.