Reference to understand this branch cut question

Question

I am currently reading a physics paper in which the authors have complexified an ordinary differential equation (ODE). They mention the following statement in the paper:

"These branch points reflect the fact that two linearly independent solutions of the wave equation will mix when they are transported around a non-trivial loop in the complex radial plane. "

I believe this is a straightforward complex analysis problem, but I have not been able to find an appropriate reference to help me fully grasp the concept. If anyone could provide a suitable reference to better understand this problem, I would greatly appreciate it.

I could refine my question a bit. Suppose we have two independent solutions given by $(x-1)^{i\alpha}$ and $(x-1)^{-i\alpha}$. When I transport them around $x=1$, I don't observe any mixing.

However, if I choose these independent solutions as $a_1 (x-1)^{i \alpha}+a_2 (x-1)^{-i \alpha}$ and $b_1 (x-1)^{i \alpha}+b_2 (x-1)^{-i \alpha}$, the first basis transforms into $a_1 e^{2\pi \alpha} (x-1)^{i\alpha}+a_2 e^{-2\pi \alpha}(x-1)^{-i\alpha}$ when I transport it around $x=1$. In this case, I can argue that the two bases mix together.

Is this a valid interpretation of the problem, and if so, can you provide any references to help me understand the underlying concepts?

Answer 1

It just means that when you have a complex-valued function $f(z)$ with a branch point (and therefore a branch cut) somewhere, the function is single-valued only as long as you do not have a contour that traverses a loop around it. As soon as you complete a full loop of $f(z)$ around a branch point, the function becomes discontinuous (or multi-valued). That's why we define branch cuts which provide artificial barriers that shouldn't be crossed. In the context you are referring to, a complex-valued solution to a differential equation will no longer remain the same if you traverse a loop around a branch point in the complex plane.