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The actual problem is to define a principal branch for $a^z=e^{zlog(a)}$ and to give a general formula. From what I understand about principal branches, it is already in that form? I'm missing something key about that. But I also am having a hard time with the general formula. The transition from real numbers to complex is confusing me, since "log" is just defined as the inverse of $e^x$, and $e^x$ is defined by the power series. I'm just not sure where to begin.

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    This helps me understand it better: as you said, $e^z$ and $logz$ are not global inverses of each other. But they can be made into _local_ inverses of each other. You can tile $C$, the complex plane by horizontal strips of length $2\mathbb pi^{-}$, i.e., half-open strips going from y to y+i2$\mathbb pi$. You can then define an invertible map between one of these strips and a copy of the complex plane with a line segment removed. Any of these maps is a branch of the log.2011-10-21
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    Basically, you can then define a map $e^z$ between the complex plane with the line with constant angle y+i2npi removed and the half open strip from y+i2(n-1)pi and y+2npi in the real plane, whose inverse image is logz. As an example, the main log, Logz is the map e^z between: the plane with the negative real axis removed, and the strip (0,2pi] in the complex plane. The inverse of this restricted map e^z is then Logz.2011-10-21
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    @gary: Why all that in two long comments? It looks pretty much like an answer to me.2011-10-21
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    Thanks for the message, joriki, I will post it.2011-10-21

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