1. Move the constants a and b out of the integral by a substitution of variables. 2. Now you've reduced the problem to calculating integrals of the form
I[n] = Int[0,INFINITY] r^n exp(-r^2) dr. Partial integration now gives you the recursion relation I[n] = (n-1)/2 * I[n-2], valid for n > 0. 3. Now you've reduced the problem to calculating I[0] and I[1]. This is most easily calculated by studying the double integrals I[0]^2 = Int[0,INFINITY] Int[0,INFINITY] exp(-x^2-y^2) dx dy = Int[0,2*pi] Int[0,INFINITY] r exp(-r^2) dr dtheta = pi and I[1] = Int[0,INFINITY] r exp(-r^2) dr = 1/2. 4. Thus, we have I[n] = 1/2 * Gamma((n+1)/2). Erik ----- Original Message ----- From: "ZHANG Yan" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Monday, May 24, 2004 2:13 PM Subject: [edstat] How to compute the integral?Thanks > Suppose that n is positive integer. Suppose that a and b are positive > real number. > > How to simplify/compute the following integral. > > Int[0, INFINITE] x^n exp^( - (x-a)^2/ (2b^2)) dx > > In this expression, "Int[0 , INFINITE]" represents the integral from > zero to infinite. x is the variable. > > Thanks for your suggestions.Any suggestions are greatly appreciated. > > -------------- > ZHANG Yan > . > . > ================================================================= > Instructions for joining and leaving this list, remarks about the > problem of INAPPROPRIATE MESSAGES, and archives are available at: > . http://jse.stat.ncsu.edu/ . > ================================================================= > . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
