Theorem on residue of composite function

Question

The following interesting and useful statement was reported some time ago on MSE:

$$ \underset{z=g(a)}{\operatorname{Res}}f(z)=\underset{z=a}{\operatorname{Res}}f(g(z))g'(z),\tag1 $$ provided that $g(z)$ is analytic in the neighborhood of $a$ and $g'(a)\ne0$.

I looked through several textbooks on complex analysis but have not found any mention of the statement. Was this result ever published?

Answer 1

Sure. You will find it, for instance, on Reinhold Remmert's Theory of Complex Functions. He calls it “transformation rule for residues”. It's on chapter 13, section 1.