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I needed some help on how to solve this problem.

Integrate $$\oint_C \frac{\sin z}{4z^2-8iz}dz $$ where $C$ consists of boundaries of the squares with vertices $\pm3,\pm3i$ counterclockwise and $\pm1,\pm i$ clockwise.

My approach was like,

$$\oint_C \frac{\sin z}{4z(z-2i)}$$ $$= \oint_C \frac{\frac{\sin z}{4z}}{z-2i}$$

So if $\frac{\sin z}{4z}$ in analytic on C, then we can use Cauchy's Integral Formula.

But, how can I check whether $\frac{\sin z}{4z}$ is analytic or not?

Marco Cantarini
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therewasaduck
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1 Answers1

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Since $\sin z$ has a root at $z=0$ and is analytic, it follows that $\frac{\sin z}{z}$ is also analytic.

More concretely, it is given by the series $$ \frac{\sin z}{z}=\sum_{n\geq 0}(-1)^n\frac{z^{2n}}{(2n+1)!} $$ and is therefore analytic. Since $\sin z$ is entire, we also see that $\frac{\sin z}{z}$ is entire (the sum converges absolutely for all $z\in \mathbb C$).

pre-kidney
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