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

\begin{eqnarray*} f(z) = z^c&= &\exp\left(c \log z\right) = \exp\left[c \left( \text{Log}\,z + 2n\pi i \right)\right], \end{eqnarray*}

with $n\in \mathbb Z.$

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.

Phase Modulus

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.