# The Complex Power Function

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

On the other hand, we have that the *generalized exponential function*, for $c \neq 0 ,$
is defined as:

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.