What is the Laurent series of the complex absolute value?

Question

What is the Laurent series of the function $f(z) = |z|$? It seems to be ill defined at $z=0$. Are there any other expansion techniques applicable for this function at $z=0$?

Answer 1

Your function isn't even holomorphic, nor is any non-constant real valued function. One easy way to see this is that if $f$ is holomorphic, then $$\left.\frac{d}{dt}f(z+tw)\right|_{t=0}=f'(z)w,$$ and if $f'(z)=0$ you can always pick $w$ so that $f'(z)w\notin\mathbb{R}$.

As for other expansion techniques, this is such a simple function, what need is there for any fancy expansions?

Answer 2

For Laurent series, you need holomorphicity in an annulus, but you do not have this. For suppose to the contrary that $f$ were holomorphic in some open set. Then, $f^2 = z\bar{z}$ would be holomorphic in the same open set. But if you take the partial with respect to $\bar{z}$, then you get $z$, which is not identically 0 in any open set.