In article <[EMAIL PROTECTED]>,
ZHANG Yan <[EMAIL PROTECTED]> wrote:
>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
Do the change of variables t = (x-a)/(b sqrt(2)). The integral becomes
b sqrt(2) int_{-a/(b sqrt(2))}^infinity (a+tb sqrt(2))^n exp(-t^2) dt
The polynomial (a+tb sqrt(2))^n can be written as a linear combination
of Hermite polynomials H_j(t) for j=0,...,n;
int H_0(t) exp(-x^2) dx = sqrt(pi)/2 erf(x) + C and
int H_n(t) exp(-x^2) dx = - exp(-x^2) H_{n-1}(x) + C for n > 0.
Robert Israel [EMAIL PROTECTED]
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
.
.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
. http://jse.stat.ncsu.edu/ .
=================================================================
