The issue I am currently having is that, while analyticity (and physics) indicates a certain function must have roots, fsolve is having trouble finding them. In fact, I have even found roots manually in a certain region myself, simply inputting into the function various values until I found them. However, fsolve does not seem to want to find these roots, and I believe it is a numerics issue: when I changed the digits around, for extremely low values of Digits, it would find a root (even though it was incorrect). Further, this problem arose elsewhere in the domain of interest for other values of Digits (in particular, for Digits:=5, fsolve failed in a region it had not failed before).
The region of interest is the "peak" of the output of poleR(M0, 0.935, mK), which should be somewhere around M0 = 0.95 or so. However, because fsolve cannot find the roots, the plot cannot be made.
Anyone have any ideas as to why fsolve cannot find the roots? I was also experiencing issues with some of these functions having multiple roots, which itself is weird as well (note that I am working over the complex plane).
Attached is the document.