| ▲ | jcranmer 3 months ago |
| The problem is a^b := exp(b ln(a)) sort of breaks down when a is negative, which is a case that is covered in algebra class but glossed over in calculus. |
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| ▲ | 3 months ago | parent | next [-] |
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| ▲ | dawnofdusk 3 months ago | parent | prev [-] |
| 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. |
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| ▲ | ogogmad 2 months ago | parent [-] | | No, 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_associat... | | |
| ▲ | dawnofdusk 2 months ago | parent [-] | | You 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). |
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