Complex Analysis

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], \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.

Phase    Modulus

Sorry, the applet is not supported for small screens. Rotate your device to landscape. Or resize your window so it's more wide than tall.

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.

NEXT: Curves in the Complex Plane

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ISBN: 978-0-6485736-0-9
© Juan Carlos Ponce Campuzano 2019-