How do I find the Laurent series expansion?

Question

I want to find the order of the pole of $\sqrt{1+\frac{1}{z}}$ at 0. But how can I expand this function at 0? Can I take the Taylor expansion of $\sqrt{1+z}$ and plug in $\frac{1}{z}$ in the place of z? And how can one find the Laurent series expansion of a function in general? I mean since the function diverges toward infinity at poles, how can the Laurent series be "defined"?

Answer 1

You're quite right that to find the Laurent series expansion for $f(z)=\sqrt{1+\frac1z}$ about $0,$ we need only start with the Taylor series expansion of $g(z)=\sqrt{1+z}$ about $0,$ then note that $f(z)=g(1/z).$ In fact, this will show you that it isn't a pole, but rather an essential singularity.

As for how the Laurent series can be thought of as being defined, recall that a power series is defined wherever it converges. In this case, it converges at all values of $z$ such that $|z|>1.$