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**arbolis** I must find the singularities and classify them (poles, essential or removable singularities) and find their principal part.

The function is $\displaystyle f(z)=\frac{\sin z}{z^2(z-\pi)}$.

My attempt :

By a first look, I notice that the function has a pole or singularity at $\displaystyle z=0$ and $\displaystyle z=\pi$.

Let's examinate what happens when $\displaystyle z=0$.

The Taylor series of $\displaystyle \sin z$ in $\displaystyle z=0$ is $\displaystyle \sum _{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!}$.

Thus $\displaystyle f(z)=\frac{1}{z(z-\pi)}-\frac{z}{z-\pi}+\frac{z^3}{z-\pi}-\frac{z^5}{z-\pi}+... $.

Now I'm unsure of myself. I see that the first term is undetermined when $\displaystyle z=0$ and $\displaystyle z=\pi$ while all the other terms are undetermined only when $\displaystyle z=\pi$. As a consequence I'm tempted to say that $\displaystyle z=0$ is a pole of order $\displaystyle 1$. (Isn't a removable singularity also?).

$\displaystyle \pi$ would be an essential singularity since there are infinite terms where $\displaystyle z=\pi$ is not defined, but I don't know how to justify formally.

Plus, another doubt I have is why did I find the Taylor series of $\displaystyle f$ in $\displaystyle z=0$? I could have done it for an arbitrary $\displaystyle z_0$, or not?