Simple way for evaluating a line integral

Question

We have a complex number c, a positive number r and a curve $\partial \Delta_r(c)$ on $\mathbb{C}$, which is given by $t \mapsto c + re^{it}$, $t \in [0, 2 \pi]$. Let $f$ be a holomorphic function. We now want to calculate the line integral \begin{equation*} \int_{\partial \Delta_r(c)} \frac{f(z)}{z(z-3i)} dz \end{equation*}

Tasks of this type were given as voluntary preparatory tasks for an exam in complex analysis at my university, and I feel that there should be a trick for solving them. My approach would be to check if $\frac{f(z)}{z(z-3i)}$ is holomorphic since it would then follow that the integral is 0, but this leads to very long and complicated equations. Is there an simpler solution?

Answer 1

It is clear tha your function is holomorphic: it is the quotient of two holomorphic functions. The integral is easy to compute using Cauchy's integral formula, but the answer depends on $r$ and $c$.

Answer 2

First observe that you are integrating in a circle with center $c$ and radius $r$ then it depends on the specific circle and whether $0,3i$ lie in it or not. If none of them are inside the circle then the integral is $0$ otherwise use Cauchys’ integral formula.