How to perform Fourier transform of a multivalued function?

[hiyok]

Hi,

There is the following function whose Fourier transform I cannot work out despite days of labour,
$$f(q) = \frac{e^{i\sqrt{q^2+1}a}}{\sqrt{1+q^2}}.$$ Here $a$ is a nonnegative constant. As usual, the Fourier transform is
$$F(x) = \int^{\infty}_{-\infty}dq~e^{iqx}f(q).$$ I tried to use contour integral, but the integrand has branch points in the complex plane. I could not find a proper contour which can make a de tour off the branch cuts!

Could you give me some advice?

Thank you !

hiyok

 

[jvicens]

oi

Hello hiyokoi,

I understand your frustration with trying to find the Fourier transform of this function. It is true that the integrand has branch points in the complex plane, which can make the calculation more challenging. However, there are a few techniques that can help with this problem.

One approach is to use the residue theorem. This involves finding the poles of the integrand and using them to evaluate the integral. In this case, the poles of the integrand are located at $q = \pm i\sqrt{1-a^2}$. By using the residue theorem, you can find the value of the integral without having to calculate it directly.

Another approach is to use the method of steepest descent. This involves deforming the contour of integration in a way that avoids the branch points and allows for an easier evaluation of the integral. This technique is often used in cases where the integrand has oscillatory behavior, as is the case with your function.

I would also recommend consulting a textbook or online resources for more specific guidance on calculating integrals with branch points. With some practice and perseverance, I am confident that you will be able to find the Fourier transform of this function. Good luck!