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Topic: Gaussian Integral Using Laplace Transforms (Read 2565 times) |
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william wu
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Gaussian Integral Using Laplace Transforms
« on: Apr 9th, 2004, 10:44pm » |
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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: Define a function f(t) = [int]0 to +inf exp(-tx2). Consider L[f(t)].
Step 2: Use the result L[1/(sqrt(pi*t)] = 1/sqrt(s).
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| « Last Edit: Apr 9th, 2004, 10:45pm by william wu » |
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Eigenray
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Re: Gaussian Integral Using Laplace Transforms
« Reply #1 on: Dec 10th, 2005, 3:37pm » |
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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 ...
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