How do I solve this integral with a branch point at z =0?

Question

The integral $\int_{-\infty}^{\infty}e^{\iota\left(k+\iota\delta\right)x^{2}}dx$ can be written as $\int_{-\infty}^{\infty}\frac{e^{\iota\left(k+\iota\delta\right)z}}{\sqrt{z}}dz$. Here, the branch point at zero is also a singularity, right ? How do I evaluate this integral ? Here, $\delta$ is positive small $0^{+}$.

Answer 1

We can actually solve the original integral given without the need of $x^2\mapsto z$. Let $u=z\sqrt{\delta - ik}$. Then $du = \sqrt{\delta - ik}dz$. $$ \int_{-\infty}^{\infty}\exp\bigl[i(k+i\delta)z^2\bigr]dz=\frac{1}{\sqrt{\delta - ik}}\int_{-\infty}^{\infty}e^{-u^2}du = \frac{\sqrt{\pi}}{\sqrt{\delta - ik}} $$ Now in order for the original integral to converge, $i(k+i\delta)<0$. By multiplying both sides by $-i$, we have $$ k+i\delta > 0\Rightarrow\Re\{k\}+\Im\{\delta\}>0 $$