I have some trouble with analytic continuation, even for a simple question. For example, I want to analytically continue the following function from negative real axis to the whole plane, $$\frac{1}{\sqrt{|k|}}\exp \Big[-\int_x^0|k|dx\Big]$$ where $k=\alpha\sqrt{x}$. The answer is $$\frac{e^{i\pi/4}}{\sqrt{\alpha\rho^{\frac{1}{2}}e^{i\phi/2}}} \exp \Big[-i\frac{2}{3}\alpha\rho^{\frac{3}{2}}e^{i3\phi/2}\Big]$$ where $\rho$ is modula and $\phi$ is the argument for $x$ when we do analytical continuation. I know for $\phi=\pi$, the second expression goes back to the first one. I want to know how to find the second one, as detailed as possible, thank you.
Let me say what I have tried. First I write the function on negative axis as $$\frac{1}{\sqrt{|k|}}\exp \Big[-\int_x^0|k|dx\Big] =\frac{1}{\alpha^{\frac{1}{2}}(-x)^{\frac{1}{4}}} \exp\Big[-\frac{2}{3}\alpha(-x)^{\frac{3}{2}}\Big] $$ Then I try to do analytical continuation. Naively if I replace $$x\rightarrow z=\rho e^{i\phi} \,\,\,\hbox{and}\,\,\,-1=e^{i\pi}$$ I got the wrong answer. Analytical continuation $$-x\rightarrow \rho e^{i(\phi-\pi)}$$ gives the right answer. I don't know why we should do the analytical continuation of $-x$ as a whole.