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- Homework Statement
- Prove that $$\int_{-\infty}^{\infty} \frac{e^{-i \alpha x}}{(x-a)^2+b^2 }dx=(pi/b) e^{-i \alpha a}e^{-b |a|}$$ via contour integration

- Relevant Equations
- Cauchy's integral $$\int_{C} \frac{f(z)}{z-z_0} dz= 2 \pi i f(z_0)$$ given C a closed curve and f(z) analytic over this curve.

Residue theorem might be useful.

$$\int_{-\infty}^{\infty} \frac{e^{-i \alpha x}}{(x-a)^2+b^2}dx=(\pi/b) e^{-i \alpha a}e^{-b |a|}$$

So...this problem is important in wave propagation physics, I'm reading a book about it and it caught me by surprise.

The generalized complex integral would be

$$\int_{C} \frac{e^{-i \alpha z}}{(z-a)^2+b^2}dz$$

I am having trouble with several things, one of them is how to define the actual curve, I imagined a conterclockwise semicircle with a keyhole on the complex pole a+ib (from -R to -epsi, upwards from -epsi to ib, semicircle from -epsi + ib to epsi +ib, downwards from epsi+ib to epsi, epsi to R, and another semicircle connecting R and -R) , I haven't been able to find much info on curves around complex poles.

Second problem I am having is that even for the first segment, I cannot for the life of me solve this integral, I tried some help with a computer and the expression involves more integrals, I do not know (and I don't think) it has an antiderivative.

Maybe there's some manipulation that I can do to take advantage of the Cauchy-Goursat theorem? maybe a variable change t=z-a might make this possible. (But given the exponential maybe one of the residues will not be zero...)

I am not looking for a full answer, but rather a good direction to take this problem to...I've been out of tune with these techniques for a while.

Thanks for your attention

So...this problem is important in wave propagation physics, I'm reading a book about it and it caught me by surprise.

The generalized complex integral would be

$$\int_{C} \frac{e^{-i \alpha z}}{(z-a)^2+b^2}dz$$

I am having trouble with several things, one of them is how to define the actual curve, I imagined a conterclockwise semicircle with a keyhole on the complex pole a+ib (from -R to -epsi, upwards from -epsi to ib, semicircle from -epsi + ib to epsi +ib, downwards from epsi+ib to epsi, epsi to R, and another semicircle connecting R and -R) , I haven't been able to find much info on curves around complex poles.

Second problem I am having is that even for the first segment, I cannot for the life of me solve this integral, I tried some help with a computer and the expression involves more integrals, I do not know (and I don't think) it has an antiderivative.

Maybe there's some manipulation that I can do to take advantage of the Cauchy-Goursat theorem? maybe a variable change t=z-a might make this possible. (But given the exponential maybe one of the residues will not be zero...)

I am not looking for a full answer, but rather a good direction to take this problem to...I've been out of tune with these techniques for a while.

Thanks for your attention