# How to perform Fourier transform of a multivalued function?

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1. May 22, 2015

### 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

2. May 27, 2015