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So we see that the Gamma function is a generalization of the factorial function. Since at integer points, the value of the Gamma function is given by the factorial, it follows that the Gamma function grows to infinity even faster than exponentials. Definition. For x,y positive we define the Beta function by. Using the substitution u = 1 - t it is easy to see that.
On the other hand, it is quite easy to show that the improper integral is convergent at regardless of the value of x. So the domain of f(x) is . which explains somehow the awkward term x-1 in the power of t. Now the domain of this new function (called the Gamma Function) is .