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It doesn't break down, one just needs the complex logarithm. If you ignore complex numbers it breaks down in both cases. If you allow complex numbers it works i
by dawnofdusk 1y ago
It doesn't break down, one just needs the complex logarithm. If you ignore complex numbers it breaks down in both cases. If you allow complex numbers it works in both cases.
- ogogmad 1y agoNo, using complex numbers alone DOES NOT work. To really allow complex numbers, you also need Riemann surfaces. The function "ln" has type ln: R -> CC where "R" denotes the Riemann surface corresponding to the natural domain of ln, and "CC" denotes the complex numbers. See here for details: https://en.wikipedia.org/wiki/Complex_logarithm#The_associated_Riemann_surface https://en.wikipedia.org/wiki/Complex_logarithm#The_associat...
- dawnofdusk 1y agoYou can also allow it to be multi-valued and consider a principal branch when needed, the same we do when we discuss roots of monomials in algebra. The two situations are identical (as they must be, because logarithms generalize roots).