Integration in complex analysis does not always seem as obvious at in the real plane, with reasons that are not obvious to me. In particular, consider $$\int_{- \infty}^{+ \infty} e^{- \pi x^2 z} dx$$
where $z$ is a complex number of positive real part, so that the integral converges. The value of this integral is $$z^{-1/2} \int_{- \infty}^{+ \infty} e^{-\pi x^2} dx = z^{-1/2}$$
and that seems to be like a standard change of variables as it would be for $z$ real. However, it is not and the argument I read for computing this integral involves Cauchy's contour theorem and turning the path of integration by an angle of $-arg(z)/2$. I do not understand:
- why isn't the change of variable (which is a "linear" one) possible, even if $z$ is complex ?
- how does the contour changing work ? (I understand it in the case of horizontal translation, but changing the angle could create convergence issues, isn't it?)
Actually, there's another way to prove this. Suppose you have a formula for an integral, say $$F(z) = \int_{-\infty}^\infty f(x,z)\; dx $$ which is true for $z$ in some interval $J$, and there is a connected open subset $U$ of
$\mathbb C$ containing $J$ such that both $F(z)$ and $\int_{-\infty}^\infty f(x,z)\;dx$ are analytic in $U$. Then the formula must be true throughout $U$.
In your example, $z^{-1/2}$ is analytic in the right half plane, while the integral is also analytic there because of locally uniform absolute convergence.