Integration by substitution involving complex numbers

Question

Say I would like to integrate e$^($$^i$$^a$$^-$$^b$$^)$$^x$, from 0 to infinity, where $a$ and $b$ are positive, real numbers. Am I allowed to substitute $x'=(ia-b)x$?

Answer 1

You seem to want to use $x$ to mean two different things. What you can say is that, for any fixed $z\in\mathbb{C}$, $\frac{dy}{dx}=zy\iff y(x)=y(0)e^{zx}$. You can also say that $\int_0^\infty e^{-zx}dx=\frac{1}{z}$ provided $\Re z > 0$.