Fourier transform of $\exp{\left(irx/\sqrt{1-ix}\right)}/(1-ix)$.

Question

I want to calculate the following Fourier transform:

$$u(r,t) = \int_{-\infty}^{\infty}d\omega\ \frac{1}{1-i\omega}\, \exp\left\{\frac{ir \omega}{\sqrt{1-i \omega}} +i\omega t\right\} $$

Initially I tried using residues, but I couldn't find a way to represent this as a meromorphic function. Any ideas on how to do this? Even a series representation would do at this point.

Answer 1

The command of Mathematica

FourierTransform[Exp[I*r*x]/Sqrt[1 - I*x]/(1 - I*x), x, s,  Assumptions -> r > 0]

results in $\frac{e^{-| r+s| } \left(| r+s| \left(-i e^{2 | r+s| } \text{erf}\left(\sqrt{| r+s| }\right)+i \text{erfi}\left(\sqrt{| r+s| }\right)+(1+i) \left(e^{2 | r+s| }-1\right) C\left(\frac{(1+i) \sqrt{| r+s| }}{\sqrt{\pi }}\right)-(1-i) \left(e^{2 | r+s| }+1\right) S\left(\frac{(1+i) \sqrt{| r+s| }}{\sqrt{\pi }}\right)+2\right)+(r+s) \left((1+i) \left(e^{2 | r+s| }+1\right) C\left(\frac{(1+i) \sqrt{| r+s| }}{\sqrt{\pi }}\right)-i \left(e^{2 | r+s| } \text{erf}\left(\sqrt{| r+s| }\right)+\text{erfi}\left(\sqrt{| r+s| }\right)-(1+i) \left(e^{2 | r+s| }-1\right) S\left(\frac{(1+i) \sqrt{| r+s| }}{\sqrt{\pi }}\right)-2 i\right)\right)\right)}{\sqrt{2} \sqrt{| r+s| }}.$

Addition. The command of Maple

inttrans:-fourier(exp(r*x*I)/(sqrt(1 - x*I)*(1 - x*I)), x, s) assuming r>0

performs $$ 4 \sqrt{\pi}\, \sqrt{s-r}\, {\mathrm e}^{-s+r} \mathrm{Heaviside}\! \left(s-r\right).$$