Deriving the Gamma function?

Anonymous

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For some reason the last time I posted this question it got removed, which I find odd. I'm in CEGEP (quebec college) and I'm trying to do a little extra learning on the side. Can somebody show me how the Gamma function (for the analytical continuation of the factorial function) is derived or point me to a site with a complete derivation. I've tried looking it up online, but every site simply states the function and then works backwards to show that it represents a factorial.

 

Answer No.1

Anonymous

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Consider this integral: ∫ [0,∞] x^ne^(-x) dx where n is a positive integer. it is not hard, and kind of interesting, to prove that this equals n! (for a proof, look here:) http://answers.yahoo.com/question/index;… this has to do with the relationship between x^n and e^x that you get when you do integration by parts, and the fact that: ∫ [0,∞] xe^(-x) dx = 1 however, x^n can be defined as a function of n, which doesn't HAVE to be a natural number. that is x^y for any real y still makes sense. and while there is no natural way to extend the factorial function f(n) = n! in any obvious way for non-integers, replacing n by y and defining f(y) = ∫ [0,∞] x^ye^(-x) dx does lead to a well-defined function of y, as long as the integral converges. there is a problem with the integration when y = -1, since at 0 the function is unbounded. mathematicians, peculiar creatures that they are, like to move such difficulties to 0, where problems are often expected. so instead they define: f(y) = ∫ [0,∞] x^(y-1)e^(-x) dx since for y < 0, integration by parts will eventually make this reduce to an integral of the form: ∫ [0,∞] x^(y-1+n)e^(-x) dx where y-1+n > 0, as long as we avoid y = 0,-1,-2,-3,.... we wind up with a well-defined function f, with f(n) = (n-1)! for any positive integer n. i'd like to emphasize that the gamma function, so defined, is not the ONLY way to make a continuous extension of the factorial function, but it does have a certain simplicity. and the exponential function and power functions are well-defined, and have nice properties (including generalizing to complex numbers in a nice algebraic way).Good

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