The *generalized complex power function* is defined as:
\begin{eqnarray}\label{gcp}
f(z) = z^c = \exp(c\log z), \quad \text{with}\quad z\neq 0.
\end{eqnarray}

Due to the multi-valued nature of $\log z$, it follows that (\ref{gcp}) is also multi-valued for any non-integer value of $c$, with a branch point at $z=0$. In other words \begin{eqnarray*} f(z) = z^c&= &\exp\left(c \log z\right) = \exp\left[c \left( \text{Log}\,z + 2n\pi i \right)\right], \text{ with } n\in \mathbb Z. %&= & \exp\left(c \,\text{Log}\,z\right)\cdot \exp\left(2 n \pi c\, i\right) \\ %&= & \exp\left[ c \left( \ln |z| +i\, \textbf{Arg} \left(z\right)\right) \right] \cdot \exp\left(2 n \pi c\, i\right)\\ %&= & |z|^c \cdot \exp\left[ c\left(\textbf{Arg} (z) + 2n\pi \right) i\right], \quad\text{ with } n\in \mathbb Z. \end{eqnarray*}

On the other hand, we have that the *generalized exponential function*, for $c \neq 0 $, is defined as:
\begin{eqnarray}\label{gef}
f(z)=c^z=\exp(z\log c)=\exp\left[z \left(\text{Log}\,c +2 n \pi \, i\right)\right],
\end{eqnarray}
with $n\in \mathbb Z$.

Notice that (\ref{gef}) possesses no branch point (or any other type of singularity) in the infinite complex $z$-plane. Thus, we can regard the equation (\ref{gef}) as defining a set of independent single-value functions for each value of $n$.

This is the reason why the multi-valued *nature* of the function $f(z)=z^c$ differs from the multi-valued function $f(z)=c^z$.

Typically, the $n=0$ case is the most useful, in which case, we would simply define: $$w=c^z=\exp(z\log c)=\exp(z\,\text{Log}\,c),$$ with $c\neq 0$.

This conforms with the definition of exponential function $$e^z=e^x(\cos y +i\sin y )$$ where $c = e$ (the Euler constant).

Use the following applet to explore functions (\ref{gcp}) and (\ref{gef}) defined on the region $[-3,3]\times[-3,3]$. The enhanced phase portrait is used with contour lines of modulus and phase. Drag the points to change the value of $c$ in each case. You can also deactivate the contour lines, if you want.

**Final remark: **In practice, many textbooks treat the *generalized exponential function* as a single-valued function, $c^z=\exp(z\,\text{Log } c )$, only
when $c$ is a positive real number. For any other value of $c$, the multi-valued function $c^z=\exp(z \log c )$ is preferred.

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