How to determine all points for which a function f(z) is holomorphic?

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How can one find all of the points of $f(z)$ s.t. $f(z)=z^4-\frac{1}{z}$ is holomorphic. I tried using the Cauchy-riemann equations , but these are kinda tricky because of the $z^4$ term. Is there a more efficient way ?

I believe from what I've been told that this function is holomorphic on a disk centred at 2 of radius 3/2 but I don't understand how that was figured out ?