lec11-3 - The factorial function We can easily find for...

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Unformatted text preview: The factorial function We can easily find for > 0 the integral, Z e- x dx =- 1 e- x | = 1 We recall that if we take derivatives with respect to , we can interchange the order of differentiation and integration, and we find Z x n e- x dx = n ! n +1 For = 1, this gives us another definition of factorials, n ! = Z x n e x dx We even have a definition for 0!, in particular 0! = Z e- x dx = 1 Gamma functions, recursion relation We can define the function ( p ), where p does not have to be integer, ( p ) = Z x p- 1 e- x dx From the last slide, we know this is related to the factorials when p = n is an integer ( n + 1) = Z x n e- x dx = n ! So for non integer p , with p >- 1, we use factorial notation (but careful because p is not an integer!) ( p + 1) = Z x p e- x = p ! Recursion relation, continued Integrate by parts, dx p e- x dx = ( px p- 1- x p ) e- x , then ( p + 1) = Z x p e- x =- x p...
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.

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lec11-3 - The factorial function We can easily find for...

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