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Problem: Calculate the residue of the function $f(z) = \frac{z^3}{(z-1)(z^4+2)}$ at $z=0$

I am confused on where to even start on this question since there is no pole at $z=0$. Is $z=0$ even a singularity of this function?

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  • $\begingroup$ Suppose you were a mathematician in the early 19th century and you had defined the concept of a residue at the poles of meromorphic functions. How would you have generalized your definition in a consistent way such that it also applies to points where the function doesn't have a pole? $\endgroup$ – Count Iblis Jul 24 '18 at 22:07
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    $\begingroup$ @CountIblis I am guessing your wanting me to notice that if a function $f$ doesn't have a pole at $z_0$, then the coefficient $a_{-1}$ in the Laurent expansion $f(z) = \sum_{n=-\infty}^{\infty} a_n(z - z_0)^n$ is always going to be zero? $\endgroup$ – Pseudo Professor Jul 24 '18 at 22:17
  • $\begingroup$ That's right! :) $\endgroup$ – Count Iblis Jul 24 '18 at 23:12
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No, $0$ is not a singularity of $f$. Therefore, the residue at $0$ is $0$.

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