Contour Choosing Problem

[Mitra]

I would like to find this integral analytically by using “Cauchy Residue Theorem”, but my problem is what contour is suitable?

$$\int {\frac {\exp(-M\omega) \exp(iN\omega)} {\omega^5(\frac{\omega^4}{f^4} + \frac{(\omega^2)(4\zeta^2-2)}{f^2}+ 1)}d\omega$$

From 0 to $$\infty$$
Where, M, N,f and $$\zeta$$ are real and positive.

Since the function has a singularity at 0 on the Semi circle Contour, what do you suggest as a contour? A Keyhole Contour? or a semicircle contour with a small detour around 0?

 

[Aaron Barnard]

For this integral, a keyhole contour would be the most suitable. The keyhole contour is a closed loop path in the complex plane that follows the real axis from 0 to some positive value M, encircles the origin in a small circle of radius r, and then returns to the real axis at M. This contour avoids the singularity at the origin and allows us to apply Cauchy’s theorem to evaluate the integral.

 

[tacman]

The choice of contour is crucial in evaluating integrals using the Cauchy Residue Theorem. In this case, we have a rational function with a singularity at 0, which means we cannot use a simple closed contour such as a semicircle or a keyhole. We need to modify the contour to avoid the singularity at 0 while still enclosing all the poles of the function.

One possible approach is to use a semicircle contour with a small detour around 0, as you suggested. This detour can be chosen to be a small semicircle centered at 0, with radius r. This way, we can avoid the singularity at 0 while still enclosing all the poles of the function.

Another approach could be to use a keyhole contour, where the detour around 0 is a small circle centered at 0, with radius r. This contour would also avoid the singularity at 0 and enclose all the poles, but it may require more calculations to determine the residues at the poles.

Ultimately, the choice of contour will depend on the specific problem and the poles of the function. It may be helpful to graph the function and its poles to get a better understanding of the contour that would be suitable. Whichever contour you choose, make sure to carefully calculate the residues at the poles to accurately evaluate the integral using the Cauchy Residue Theorem.