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$f(z) = \frac{cosz}{sin(z^2)(e^z -1)}.$

Find the principal part of the Laurent series relative to $f(z)$.


I have found the principal part of $f(z)$ for $z=0$, but there is another pole at $z=\sqrt{k\pi}$, and I have no idea how to proceed.

For $z=0$, I used geometric series to find a series expansion for $1/sin(z^2)$ and $1/(e^z -1)$ and multiplied them with the expansion of $cos z$. Can I use a similar method for $z=\sqrt{k\pi}$?

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