- #1

Santilopez10

- 81

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Hello! I have been searching the web and textbooks for a certain theorem which generalizes the value of the integral around a infinitesimal contour in the real axis, or also called indented contour over a nth order pole.

It is easy to prove that if the pole is of simple order, the value of the integral is just ##-i\pi Residue(f(z_0))##, but problem arises when ##n\geq2##. for example: $$ \oint_C \frac {e^{iz(x-1)}}{z^2}\, dz $$

Which arises in the search for the inverse Fourier transform of the Fourier transform of x, in the range 0<x<1, which basically is proving the duality of the Fourier transform for the defined function.

Now, if our contour has a indented path at z=0, we make the substitution ##z=\epsilon e^{i \phi}, dz=i \epsilon e^{i \phi} d\phi## and we let ##\phi## run from ##\pi## to 0, we get: $$ \int_\pi^0 \frac {e^{i\epsilon e^{i\phi} (x-1)}}{\epsilon^2 e^{2i\phi}} i\epsilon e^{i\phi}\, d\phi$$

which isn`t defined if we let ##\epsilon \rightarrow 0##. So I am stuck there, the integral should exist, as it does have a defined inverse Fourier transform in the range 0<x<1. So the question is: Does anyone know any theorem which states the value of the integral for higher order poles in the real axis?

EDIT: The function is square integrable, so the inverse Fourier transform must exist.

the origal Fourier transform is ## \frac {e^{-iw}(iw-e^{iw}+1)}{w^2} ##

It is easy to prove that if the pole is of simple order, the value of the integral is just ##-i\pi Residue(f(z_0))##, but problem arises when ##n\geq2##. for example: $$ \oint_C \frac {e^{iz(x-1)}}{z^2}\, dz $$

Which arises in the search for the inverse Fourier transform of the Fourier transform of x, in the range 0<x<1, which basically is proving the duality of the Fourier transform for the defined function.

Now, if our contour has a indented path at z=0, we make the substitution ##z=\epsilon e^{i \phi}, dz=i \epsilon e^{i \phi} d\phi## and we let ##\phi## run from ##\pi## to 0, we get: $$ \int_\pi^0 \frac {e^{i\epsilon e^{i\phi} (x-1)}}{\epsilon^2 e^{2i\phi}} i\epsilon e^{i\phi}\, d\phi$$

which isn`t defined if we let ##\epsilon \rightarrow 0##. So I am stuck there, the integral should exist, as it does have a defined inverse Fourier transform in the range 0<x<1. So the question is: Does anyone know any theorem which states the value of the integral for higher order poles in the real axis?

EDIT: The function is square integrable, so the inverse Fourier transform must exist.

the origal Fourier transform is ## \frac {e^{-iw}(iw-e^{iw}+1)}{w^2} ##

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