#### marvellover

##### New member

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- Thread starter marvellover
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I meant any equation that is considered gaussian in 1 D

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Please give us an example of:I meant any equation that is considered gaussian in 1 D

Gaussian equation in 1 D that you need to solve, using laplace transform and fourier transform.

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\(\displaystyle G(x) = \dfrac{1}{ \pi } e^{-(x- x_m)^2/ ( \Delta x )^2}\)

\(\displaystyle G(x, y) = \dfrac{1}{ \pi ^2 } e^{-(x- x_m)^2/ ( \Delta x )^2} ~ e^{-(y- y_m)^2/ ( \Delta y )^2}\)

This is a bit different from how the Gaussian function is usually written but you can fix that if you need to. Any Calculus methods will work the same way.

Take G(x, y) and modify it a bit, just to show the method.

\(\displaystyle G(x, y) = A e^{-x^2} e^{-y^2}\)

\(\displaystyle \int_{- \infty }^{ \infty } \int_{- \infty }^{ \infty } A e^{-x^2 - y^2 } ~ dx ~ dy\)

Convert to polar coordinates:

\(\displaystyle = \int_{0}^{ \infty } \int_{0}^{2 \pi } A e^{-r^2} ~ r ~ dr ~ d \theta \)

You can now use integration by parts.

The G(x) integration is done in exactly the same way. \(\displaystyle G(x) G(y) = G(x, y)\), so \(\displaystyle \int G(x) ~ dx = \sqrt{ \int G(x, y) dx dy}\)

-Dan

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THANK YOU SO MUCH SIR

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