I'm not studying several complex variables, but I may need to use some elementary results from the subject—particularly those regarding singularities.
I know Hartogs' Extensions Theorem boils down to: "isolated singularities of holomorphic functions of multiple variables are removable", but I'm having a difficult time wrapping my head around it.
One example that does make sense to me is $\frac{1}{z-w}$; there, the singularities are indeed non-isolated. However, what about something like $\frac{1}{z-a}\frac{1}{w-b}$ or $\frac{z-a}{w-b}$?—where $a$ and $b$ are (possibly non-distinct) complex numbers. Are the singularities of these functions truly removable? If so, how do you go about analytically continuing the functions to take values at those singularities—i.e., what are the "correct" values there? On the other hand, if they aren't removable, how do they not satisfy the hypotheses of Hartogs' Extension Theorem?