For the function [tex]f(x)[/tex] given by:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]f(x) = e^{2x} (x<0), = e^{-x} (x>0)[/tex]

I have got the complex Fourier Transform to be:

[tex] F(k) = {3(k^{2} + ik + 2)}/{(k^{2}+1)(k^{2} + 4)}[/tex]

Now I'm trying to verify the formula for the inverse transform by using a D-contour integral. Just taking the x>0 case I have found the strip of regularity and am closing the D- contour below (to avoid the exponential exploding).

Closing the contour below gives 3 poles in the contour, namely:

[tex]k=i,-i,-2i[/tex]

I have "argued away" the curve of the Dcontour okay.

So now computing the sum of the residues and multiplying by (-2pi i) should give me back:

[tex]f(x) = e^{-x}[/tex]

The residues at k= i and k= -2i are zero so just working on k= -i:

[tex]res = {3(k^{2}+ik+2)e^{-ikx}}/{4k^{3} + 10k}[/tex]

I got this by just differentiating bottom line (trick for getting formula for res)

when k = -i this gives:

[tex]res = {-2e^{-x}}/(i)[/tex]

multiplying by (-2pi i) and dividing by 2pi (according to inv transform formula) gives..

[tex]f(x) = 2e^{-x}[/tex]

why have I got that 2?!

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# Complex Fourier Transform & Its Inverse (also Dcontour integrals)

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