Determine if $f(z)$ has a branch point where it is not analtyic?

Question

Let us say that you have a function $f(z)$ how would determine if the point $z=z_0$ is a branch point of $f(z)$ (and its order) given that $f(z)$ is not analytic at this point.

(For reference: this question (and the links in the answer) Showing $1$ is not a branch point for $f(z) = z^2$? gives the case where $f(z)$ is not analytic at $z_0$)

Answer 1

$\newcommand{\Cpx}{\mathbf{C}}$Suppose $g:\Cpx \to \Cpx$ is holomorphic. Loosely, branch points of $f = g^{-1}$ correspond to critical points of $g$: If $g(w_{0}) = z_{0}$ and $g'(w_{0}) = 0$, then $f = g^{-1}$ has a branch point at $z_{0}$.

For example, the squaring function $g(w) = w^{2}$ has a critical point at $w_{0} = 0$, so the square root $f(z) = \sqrt{z}$ has a branch point at $0 = z_{0} = g(w_{0})$.

In more detail, if $g(w_{0}) = z_{0}$ and $g'(w_{0}) = 0$, then there exists a unique integer $n \geq 2$ and a unique holomorphic function $h$ with $h(w_{0}) \neq 0$ (and therefore non-vanishing in some neighborhood of $w_{0}$) such that $$ z = g(w) = z_{0} + (w - w_{0})^{n}\, h(w) $$ for all $w$ in some neighborhood of $w_{0}$. The (necessarily multi-valued) inverse function $f$ defined by $$ f(z) = w_{0} + \left[\frac{z - z_{0}}{h(w)}\right]^{1/n} $$ satisfies $f(z_{0}) = w_{0}$ and $g\bigl(f(z)\bigr) = z$ for all $z$ in some neighborhood of $z_{0}$, and has a branch point of order $n$ at $z_{0}$.

Conversely, if $g'(w_{0}) \neq 0$, the inverse function theorem guarantees $g$ is locally a biholomorphism, i.e., has a (single-valued) holomorphic inverse in some neighborhood of $z_{0} = g(w_{0})$.

Since all these considerations are local, similar remarks hold for holomorphic mappings between arbitrary Riemann surfaces.

Answer 2

Here are my steps to finding the branch points of the function $f(z)$

1. Set $w=f(z)$

2. Differentiate both sides w.r.t $w$:$$1=f'(z) \frac{dz}{dw}$$ $$\frac{dz}{dw}=\frac{1}{f'(z)}$$

3. Find the values of $z$ for which the above is $0$. These are our branch points.

   4. Set $t=\frac{1}{z}$ and use the above method to find the branch points of $f(t)$. If $t=0$ is a branch point, then our original function has a branch point at $\infty$.

To find the order of the branch point at $z=z_0$.

Method 1

Find the first time derivative is not 0, i.e. that $$\frac{d^n z}{dw^n}\ne0$$ and $$\frac{d^k z}{dw^k}=0 \forall k<n$$ (all evaluated at $z=z_0$). Then the order of this branch point is $n$.

Method 2

Sub $z=z_0+re^{i\theta}$ into the original function and physically see how many times you have to take $\theta$ round to get back to your original value. If you can take it $n+1$ times round the order is $n$.