# Are there complex functions with finite, nonzero branch points?

I just completed a brief introduction to branch points in complex analysis, and I find it difficult to imagine/come up with functions with nonzero branch points.

My difficulty is this: for the point to be considered a branch point, f(r,θ) and f(r,θ+2π) must be different for ANY closed path. If the branch point is zero, you can arbitrarily shrink the path and still be sweeping from 0 to 2π. If your branch point is anything but the origin, then as the path becomes arbitrarily small, so does the angle you sweep over. In such a case, how can you get multiple values for the same input?

WWGD
Gold Member
What if you just translate a function with branch point at 0?

FactChecker
Gold Member
f(r,θ) and f(r,θ+2π) must be different for ANY closed path.
You mean for any closed path with a winding number around zero. This is specific for a branch point at zero. For a branch point somewhere else, the closed paths to consider are those with a winding number around that other point, not around zero.

fuserofworlds: look for Riemann surfaces in textbooks or the Internet. They provide a conceptual explanation of branching functions.

jasonRF
Hint: find the branch points of $\sqrt{(z-a)(z-b)}$ for $a \neq b$ and neither $a$ or $b$ are zero.