Complex power of a complex number Can someone explain to me, step by step, how to calculate all infinite values of, say, 
$(1+i)^{3+4i}$? 
I know how to calculate the principal value, but not how to get all infinite values...and I'm not sure how to insert the portion that gives me the other infinity values.

Let's suppose you've already defined $\log r$ for real $r > 0$, say, using Taylor series. Then given $z, \alpha \in \mathbb{C}$, you can define
$$z^{\alpha} = \exp(\alpha \log z)$$
where
$$\exp(w) = \displaystyle \sum_{j=0}^{\infty} \dfrac{z^j}{j!} \qquad \text{and} \qquad \log(w) = \log |w| + i \arg(w)$$
This is not well-defined - it relies on a choice of argument, which is well-defined only up to adding multiples of $2\pi$. It's these multiples of $2\pi$ which give you new values of $z^{\alpha}$.
Explicitly, if $w$ is one value of $z^{\alpha}$, then so is
$$w \cdot e^{2n \pi \alpha i}$$
for any $n \in \mathbb{Z}$.
Fun facts ensue:


*

*if $\alpha$ is an integer then $z^{\alpha}$ is well-defined

*if $\alpha$ is rational then $z^{\alpha}$ has finitely many values

*if $\alpha$ is pure imaginary then $z^{\alpha}$ is real (but not well-defined)

