- #1
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Dear all,
I have recently come across the following Fourier transform (FT):
[itex] <br /> I=\int_{-\infty}^{\infty} dx \, e^{-\imath x t} \frac{(1-x^2)}{(1+x^2)^{3/2} (a^2+x^2)}.<br /> [/itex]
The integrand contains two branch points on the imaginary axis, plus two poles also residing on the imaginary axis. The two simple poles can either be on or off the branch cuts depending on the value of the constant "a". Admittedly, I am experiencing some problems evaluating the above FT. The integrand admit a finite analytical result (I mean when evaluated without the exponential term), so I believe also its FT should exists. One of the strategies I have tried has been to rewrite the integral as:
[itex] <br /> I=\int_{-\infty}^{\infty} dx \, \frac{e^{-\imath x t}}{(1+x^2)^{1/2}} \frac{(1-x^2)}{(1+x^2) (a^2+x^2)}.<br /> [/itex]
and use the convolution theorem for the "two" functions. The FT of the inverse square root function is the modified Bessel function of the First kind, while the second one can be easily evaluated since it only contains simple poles. I have performed the convolution using Mathematica, that after some time gave me a finite and nice looking answer. Unfortunately, by confronting the numerical and the analytical solution graphically, the two plots look slightly different, even though they show a similar behaviour... clearly something is missing.
Has everyone came across something similar? Do you have any suggestions on strategies to adopt in order to solve the above integral?
Thank you!
I have recently come across the following Fourier transform (FT):
[itex] <br /> I=\int_{-\infty}^{\infty} dx \, e^{-\imath x t} \frac{(1-x^2)}{(1+x^2)^{3/2} (a^2+x^2)}.<br /> [/itex]
The integrand contains two branch points on the imaginary axis, plus two poles also residing on the imaginary axis. The two simple poles can either be on or off the branch cuts depending on the value of the constant "a". Admittedly, I am experiencing some problems evaluating the above FT. The integrand admit a finite analytical result (I mean when evaluated without the exponential term), so I believe also its FT should exists. One of the strategies I have tried has been to rewrite the integral as:
[itex] <br /> I=\int_{-\infty}^{\infty} dx \, \frac{e^{-\imath x t}}{(1+x^2)^{1/2}} \frac{(1-x^2)}{(1+x^2) (a^2+x^2)}.<br /> [/itex]
and use the convolution theorem for the "two" functions. The FT of the inverse square root function is the modified Bessel function of the First kind, while the second one can be easily evaluated since it only contains simple poles. I have performed the convolution using Mathematica, that after some time gave me a finite and nice looking answer. Unfortunately, by confronting the numerical and the analytical solution graphically, the two plots look slightly different, even though they show a similar behaviour... clearly something is missing.
Has everyone came across something similar? Do you have any suggestions on strategies to adopt in order to solve the above integral?
Thank you!