Question: "The factorial n! of a non-negative integer n is the product of all positive integers less than or equal to n. The gamma function is an extension of the factorial function to real numbers. It is defined by:
(a) Evaluate
Assuming the limits represent t, if I evaluate , does the function become ? In which case when I integrate do I get: with limits between 0 and infinity, which equals 1?
Part (b):
"Show that for x>0:
So I thought that . If I let , then , and if I let , then . Then I can use 'by parts' to show this. Do I treat x as some constant, like n?
Finally:
"(c) Show that for n=1,2,3...
That is, the gamma function generalizes the factorial, with its argument shifted down by 1"
I am entirely unsure about how to work this one out. Any tips/hints? Many thanks!
CP