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        riddles >> putnam exam (pure math) >> Gaussian Integral Using Laplace Transforms
        
(Message started by: william wu on Apr 9^th, 2004, 10:44pm)

Title: Gaussian Integral Using Laplace Transforms
Post by william wu on Apr 9^th, 2004, 10:44pm

Many are familiar with the trick of evaluating the Gaussian integral

[int]_{-inf to inf} exp(-x² ) dx

by multiplying the integral with a copy of itself, squarerooting, and converting to polar coordinates. This problem demonstrates a neat way of evaluating the integral using Laplace Transforms. Use the hidden steps/hints below as necessary:

Step 1: [hide]Define a function f(t) = [int]_{0 to +inf} exp(-tx²). Consider L[f(t)].[/hide]

Step 2: [hide]Use the result L[1/(sqrt(pi*t)] = 1/sqrt(s).[/hide]

Title: Re: Gaussian Integral Using Laplace Transforms
Post by Eigenray on Dec 10^th, 2005, 3:37pm

But step 2 is equivalent to what you're trying to show!
L[1/sqrt(pi t)](s) = [int] 1/sqrt(pi t) e^-st dt
substituting st=x², dt = 2x/s dx,
= [int] 1/sqrt(pi x²/s) e^{-x^2} 2x/s dx
= 2/sqrt(pi s) [int] e^{-x^2} dx.

You could also prove it using the fact that
g(x)=e^{-pi x^2}
is its own Fourier transform ...