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Residue identity for function composition: $ \operatorname{Res}(f; h(a)) = \operatorname{Res}((f∘h)h'; a)$
With $a \in \mathbb{C}$, let $h: D_1 \to D_2$ and $f: D_2\setminus{\{h(a)\}} \to \mathbb{C}$ be analytic functions. Also require $h'(a) \neq 0$.
Claim: $$ \operatorname{Res}(f; h(a)) = \operatorname{Res}((f \circ h)h'; a)$$
Attempt
For $\rho$ small enough, we have:
$$ \operatorname{Res}(f; h(a)) = \frac{1}{2\pi i}\int_{uh(a)=\rho} f(u)\ \mathrm{d}u$$
and
\begin{align}\operatorname{Res}((f \circ h)h', a)& = \frac{1}{2\pi i} \int_{ua=\rho} (f \circ h) h'\ \mathrm{d}u = \frac{1}{2\pi i} \int_0^{2\pi} f(h(a + \rho e^{it})) h'(a + \rho e^{it})\rho e^{it} i\ \mathrm{d}t\\&= \frac{1}{2\pi i} \int_\alpha f(u)\ \mathrm{d}u
\end{align}
where $\alpha: [0, 2\pi] \to \mathbb{C}, \alpha(t) = h(a+\rho e^{it})$.
We see that the curve of the LHS defined by $uh(a)=\rho$ (i.e. $h(a) + \rho e^{it}$) is "approximated" by the curve defined by $\alpha$ for $\rho \to 0$ because analytic functions approximately retain circles (see here).
(Note that we already used $h'(a) \neq

Let $U \subset D_1$ be a disk with center $a$ in which $h$ is injective.
Then $V := h(U) \subset D_2$ is a simplyconnected neighbourhood of $h(a)$.
For $\rho > 0$ small enough, let
$$
\gamma: [0, 2 \pi] \to U, \gamma(t) = a + \rho e^{it}
$$
and $\alpha = h \circ \gamma$. Then – as you already computed –
$$
\operatorname{Res}((f \circ h)h', a) = \frac{1}{2 \pi i}\int_\gamma f(h(z))h'(z) \, dz = \frac{1}{2 \pi i}\int_\alpha f(w) \, dw \quad .
$$
Since $V$ is simplyconnected and $f$ is holomorphic in $V \setminus \{ h(a) \}$, we can apply the Residue theorem to
the righthand side:
$$
\frac{1}{2 \pi i}\int_\alpha f(w) \, dw = \operatorname{Res}(f, h(a)) I(\alpha, h(a))
$$
and it remains to show that the winding number $I(\alpha, h(a))$ of $\alpha$ with respect to $h(a)$ is one:
$$
I(\alpha, h(a)) = \frac{1}{2 \pi i}\int_\alpha \frac{dw}{w  h(a)}
= \frac{1}{2 \pi i}\int_\gamma \frac{h'(z)}{h(z)  h(a)} \, dz
= \operatorname{Res}(\frac{h'}{ha}, a)
$$
because $ h'/(ha)$ is holo
20180624 13:16:18