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#### skatenerd

##### Active member

- Oct 3, 2012

- 114

I'm given a Gaussian function to apply a Fourier transform to.

$$f(x)=\frac{1}{\sqrt{a\sqrt{\pi}}}e^{ik_ox}e^{-\frac{x^2}{2a^2}}$$

Not the most appetizing integral...

$$g(k)=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{a\sqrt{\pi}}}\int_{-\infty}^{\infty}e^{ik_ox}e^{-\frac{x^2}{2a^2}}e^{-ikx}dx$$

$$=\frac{1}{\sqrt{2\pi{a}\sqrt{\pi}}}\int_{-\infty}^{\infty}exp[-\frac{x^2}{2a^2}+i(k_o-k)x]dx$$

I don't really want to write out the whole process, but I multiplied this expression by

$$\frac{e^{A^2(k_o-k)^2}}{e^{A^2(k_o-k)^2}}$$

and then I found \(A\) in order to complete the square in the exponent inside the integrand,

$$A=-\frac{i\sqrt{2}a}{2}$$

So I now have the integral

$$g(k)=\frac{1}{\sqrt{2\pi{a}\sqrt{\pi}}}e^{-\frac{a^2}{2}(k_o-k)^2}\int_{-\infty}^{\infty}exp[-(\frac{x}{\sqrt{2}a}-\frac{i\sqrt{2}a}{2}(k_o-k))^2]dx$$

But this is where I get stuck. I don't really understand how to integrate this expression. I learned from my lecture that this whole completing the square process is how we simplify the integrand, but I am iffy on the process after this point. Any hints in the right direction would be nice. (Also there is definitely a chance that this integrand isn't quite correct. There is a lot of room for error to arrive at this point, though I did check my work multiple times)

$$f(x)=\frac{1}{\sqrt{a\sqrt{\pi}}}e^{ik_ox}e^{-\frac{x^2}{2a^2}}$$

Not the most appetizing integral...

$$g(k)=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{a\sqrt{\pi}}}\int_{-\infty}^{\infty}e^{ik_ox}e^{-\frac{x^2}{2a^2}}e^{-ikx}dx$$

$$=\frac{1}{\sqrt{2\pi{a}\sqrt{\pi}}}\int_{-\infty}^{\infty}exp[-\frac{x^2}{2a^2}+i(k_o-k)x]dx$$

I don't really want to write out the whole process, but I multiplied this expression by

$$\frac{e^{A^2(k_o-k)^2}}{e^{A^2(k_o-k)^2}}$$

and then I found \(A\) in order to complete the square in the exponent inside the integrand,

$$A=-\frac{i\sqrt{2}a}{2}$$

So I now have the integral

$$g(k)=\frac{1}{\sqrt{2\pi{a}\sqrt{\pi}}}e^{-\frac{a^2}{2}(k_o-k)^2}\int_{-\infty}^{\infty}exp[-(\frac{x}{\sqrt{2}a}-\frac{i\sqrt{2}a}{2}(k_o-k))^2]dx$$

But this is where I get stuck. I don't really understand how to integrate this expression. I learned from my lecture that this whole completing the square process is how we simplify the integrand, but I am iffy on the process after this point. Any hints in the right direction would be nice. (Also there is definitely a chance that this integrand isn't quite correct. There is a lot of room for error to arrive at this point, though I did check my work multiple times)

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