Gamma Function
The gamma function is the generalization of a factorial.
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Description
For a positive integer x, the function can be defined as Γ(x) = (x-1)!.
For a complex number z with a positive real part, the function can be defined as:
It should be clear that this formula evaluates to 1 when z = 1, matching the other definition (0! = 1).
Furthermore, evaluating when z = α + 1 leads to a recursive solution.
Using integration by parts with u = xa and dv = e-x dx:
Which can be succinctly rewritten as:
Again, this matches the other definition (n! = n*(n-1)!)