How to check if a primitive has a branch point

Question

Hi I am having trouble understanding how to check if a primitive function has a branch point. For example the book is telling me that considering $$ f'(z) =\frac{e^z+1}{\sin^2(iz)} $$ then $f(z)$ has a branch point in $0$ . But how can I check that without using the integral calculus? Is there a way?

Answer 1

Look at the Laurent series of $f'(z)$ around $z = 0$: $$ f'(z) = -\frac 2 {z^2} - \frac 1 z + \frac 1 6 + \dots $$ Notice that it has a non-zero $\frac 1 z$ term - this is the important point.

Now look what happens when you integrate this term by term: $$f(z) = \frac 2 z - \log z + c + \frac z 6 + \dots $$ where $c$ is a constant.

The log that appears is the source of the branch cut.