Do biholomorphic maps respect the existence of residues Let $\varphi : U \to V$ be a biholomorphism between subsets of the plane. Let $f : V \to \mathbb C$ be an analytic function, except possibly for isolated singularities. I have proved the following facts:


*

*The map $\varphi$ respects the orders of the zeros and poles of $f$.

*The map $\varphi$ respects the essential singularities of $f$.

*The map $\varphi$ need not respect the exact values of $f$'s residues.
My next question is — does $\varphi$ at least respect the existence of residues? That is, if $f$ has nonzero residue at $z_0 \in V$, will $f \circ \varphi$ also have nonzero residue at $\varphi^{-1}(z_0) \in U$? Of course, if $f$ has at most a pole of order one, the answer is obviously “yes”. But I'm interested in the general case.
 A: For poles of order $> 1$ and essential singularities, composition with a biholomorphic $\varphi$ can change nonzero residues to zero residues and vice versa.
For example, consider - on small neighbourhoods of $0$ - the biholomorphic function $\psi \colon z \mapsto z - z^2$ and the meromorphic function $g(w) = w^{-2}$. Then $\operatorname{Res}(g; 0) = 0$ and
$$f(z) = g(\psi(z)) = \frac{1}{(z - z^2)^2} = \frac{1}{z^2}\sum_{n = 0}^{\infty} (n+1)\cdot z^n = \frac{1}{z^2} + \frac{2}{z} + 3 + 4z + \dotsc$$
has $\operatorname{Res}(f; 0) = 2$. Let $\varphi = \psi^{-1}$, then $f\circ \varphi = g$ has residue $0$ at $0$.
The fact that the residue of a function is not preserved under composition with biholomorphic maps means that we cannot in general define residues of holomorphic functions at isolated singularities on Riemann surfaces. However, things change if instead of holomorphic functions with isolated singularities we look at holomorphic $1$-forms with isolated singularities. Since for a closed curve $\gamma$ and a function $f$ that is holomorphic on the trace of $\varphi \circ \gamma$ we have
$$\int_{\varphi \circ \gamma} f(z)\,dz = \int_{\gamma} f(\varphi(w))\cdot \varphi'(w)\,dw$$
the residue of the $1$-form $\omega \colon z \mapsto f(z)\,dz$ at an isolated singularity $z_0$ - which is defined as
$$\operatorname{Res}(\omega; z_0) := \frac{1}{2\pi i} \int_{\Gamma} \omega$$
where $\Gamma$ is a closed curve winding once around $z_0$ and not winding around any other (non-removable) singularity of $\omega$ - is preserved under the pull-back with biholomorphic functions, i.e. $\operatorname{Res}(\omega; \varphi(w_0)) = \operatorname{Res}(\varphi^{\ast}\omega; w_0)$, where
$$\varphi^{\ast}\omega = \omega \circ \varphi \colon w \mapsto f(\varphi(w))\,d\varphi(w) = f(\varphi(w))\varphi'(w)\,dw\,.$$
Thus we can define residues of holomorphic $1$-forms at isolated singularities and employ the residue calculus also on Riemann surfaces.
