I want to make Fourier transform of $\frac{1}{\sqrt{x}}$, so I need to calculate this Integrl:
$$ \lim_{R\rightarrow\infty}\int_{-R}^{R}\frac{\mathrm{e}^{-\mathrm{i}pz}}{\sqrt{z}}\:\mathrm{d}z $$
I use complex analysis to calculate. However, when I made secantlines at different positions on the complex plane, I got different answers. For example, when the secantline is on the lower half plane, the integral is $0$ when $p < 0$. But if the secantline is on the upper half plane, the integral is $0$ when $p > 0$, why?
First thing to do : For $x > 0$, let $u = \sqrt{x}$ so $dx = 2udu$
$$\int_{0}^R \frac{e^{-ipx}}{\sqrt{x}}dx = \int_{0}^{\sqrt{R}} 2e^{-ipu^2}du$$
For $x < 0$, with your definition of $\sqrt{z}$ we have $\sqrt{x} = i\sqrt{-x}$ and we can define $u = \sqrt{-x}$ and $dx = -2udu$
$$\int_{-R}^0 \frac{e^{-ipx}}{\sqrt{x}}dx = \int_{-R}^0 \frac{e^{-ipx}}{i\sqrt{-x}}dx = \int_{\sqrt{R}}^{0} 2ie^{ipu^2}du = -2i \int_0^{\sqrt{R}}e^{ipu^2}du$$
So we have to compute the Fresnel integral : $$\int_0^{\infty} e^{iau^2}du = \frac{1}{2}\sqrt{\dfrac{\pi}{2|a|}}(1 + i \times sgn(a))$$