Section 2.10 Logarithms of Complex Numbers
How can we extend the logarithm function to complex numbers? We would like to retain the property that the logarithm of a product is the sum of the logarithms:
\begin{equation}
\ln(ab)=\ln a+\ln b\tag{2.10.1}
\end{equation}
Then, if we write the complex number \(z\) in exponential form:
\begin{equation}
z=r\, e^{i(\theta+2\pi m)}\tag{2.10.2}
\end{equation}
and use the property (2.10.1), we find:
\begin{align}
\ln z\amp = \ln (r\, e^{i(\theta+2\pi m)})\notag\\
\amp = \ln r+ \ln (e^{i(\theta+2\pi m)})\notag\\
\amp = \ln r+ i(\theta+2\pi m)\tag{2.10.3}
\end{align}
The logarithm function (for complex numbers) is an example of a multiple-valued function. All of the multiple-values of the logarithm have the same real part \(\ln r\) and the imaginary parts all differ by \(2\pi\text{.}\)
An interesting problem to try is to find \(\ln(-1)\text{.}\) You were probably told in high school algebra that the logarithms of negative numbers do not exist.