- #1
MathematicalPhysics
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For the function [tex]f(x)[/tex] given by:
[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?!
[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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