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

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?

• 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? – Count Iblis Jul 24 '18 at 22:07
• @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? – Pseudo Professor Jul 24 '18 at 22:17
• That's right! :) – Count Iblis Jul 24 '18 at 23:12

## 1 Answer

No, $0$ is not a singularity of $f$. Therefore, the residue at $0$ is $0$.