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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.