We are supposed to calculate a few integrals where the integrand is a quotient of holomorphic functions. Wikipedia says (and there is a proof on this site) that the quotient $(f/g)$ is holomorphic wherever $g\ne 0$.
We also have the following lemma from class:
Let $G$ be a simply connected domain, $c \in G$, and let $f \in C(G)$ be holomorphic on $G \setminus \{c\}$. Then for all closed piecewise $C^1$-paths $\gamma$ in $G$ $$\int_\gamma f(z)dz = 0$$
This seems way too strong a formula (as far as I understand it). For instance the integrals
- $$\int_{\partial D} \frac{\cosh(z)}{4z^3-z}dz$$
- $$\int_{\partial D} \frac{\cos(\pi z)}{(z-2)^3}dz$$
would evaluate to zero, as long as the closed disks $D$ contain at most one of the roots of the denominator function:
- $\{-0.5,0,0.5\}$
- $\{2\}$
respectively. This root would then be our $c$, $D$ our connected domain $G$, and f would be holomorphic on $G\setminus\{c\}$.
I have a strong feeling this is wrong, as it seems too good to be true. But many properties of holomorphic functions seem too good to be true, that's why I'm asking.
Best Answer
Not quite. In order to apply that theorem in order to compute $\displaystyle\int_{\partial D}\frac{\cosh(z)}{4z^3-z}\,\mathrm dz$, the disk $D$ must contain none of the roots of $4z^3-z$. Otherwise, $\dfrac{\cosh(z)}{4z^3-z}\notin C(D)$, since, for each $z_0\in\left\{-\frac12,0,\frac12\right\}$, the limit $\displaystyle\lim_{z\to z_0}\frac{\cosh(z)}{4z^3-z}$ doesn't exist.