Complex Analysis

The Logarithmic Function


Consider $z$ any nonzero complex number. We would like to solve for $w$, the equation \begin{eqnarray}\label{log1} e^w=z. \end{eqnarray} If $\Theta=\textbf{Arg} (z)$ with $-\pi < \Theta \leq \pi$, then $z$ and $w$ can be written as follows \begin{eqnarray*} z=re^{i\Theta} \quad \text{and}\quad w=u+iv. \end{eqnarray*} Then equation (\ref{log1}) becomes $$e^ue^{iv}=re^{i\Theta}.$$ Thus, we have \begin{eqnarray*} e^u=r\quad\text{and}\quad v=\Theta +2n\pi \end{eqnarray*} where $n\in \mathbb Z $. Since $e^u=r$ is the same as $u=\ln r$, it follows that equation (\ref{log1}) is satisfied if and only if $w$ has one of the values \begin{eqnarray*} w=\ln r +i(\Theta + 2n\pi )\quad (n\in \mathbb Z ). \end{eqnarray*}

Therefore, the (multiple-valued) logarithmic function of a nonzero complex variable $z=re^{i\Theta}$ is defined by the formula \begin{eqnarray}\label{log2} \log z=\ln r +i\left(\Theta + 2n\pi \right)\quad \quad(n\in \mathbb Z ). \end{eqnarray}

Example 1: Calculate $\log z$ for $z=-1-\sqrt{3}i$.

Solution: If $z=-1-\sqrt{3}i$, then $r=2$ and $\Theta=-\frac{2\pi}{3}$. Hence $$\log(-1-\sqrt{3}i)=\ln 2 +i\left(-\frac{2\pi}{3}+2n\pi\right)=\ln 2 +2\left(n-\frac{1}{3}\right)\pi i$$ with $n\in \mathbb Z .$

The principal value of $\log z$ is the value obtained from equation (\ref{log2}) when $n = 0$ and is denoted by $\text{Log} \,z$. Thus $$\text{Log}\, z=\ln r +i\Theta .$$ The function $\text{Log}\, z$ is well defined and single-valued when $z\neq 0$ and that $$\log z=\text{Log}\, z+2n\pi i\quad (n\in \mathbb Z ) $$ This is reduced to the usual logarithm in calculus when $z$ is a positive real number.

Example 2: Calculate $\log \left(1\right)$ and $\log \left(-1\right)$.

Solution: From expression (\ref{log2}) $$\log \left(1\right)=\ln 1+i\left(0+2n\pi\right)=2n\pi i\quad \quad(n\in \mathbb Z ) $$ and $$\log \left(-1\right)=\ln 1+i\left(\pi+2n\pi\right)=\left(2n+1\right)\pi i\quad \quad(n\in \mathbb Z ) $$ Notice that $\text{Log}\, (1)=0$ and $\text{Log}\, (-1)=\pi i$.

Expression (\ref{log2}) is also equivalent to the following: \begin{eqnarray*} \log z&=&\ln |z| +i\,\textbf{arg} (z)\\ &=&\ln |z| +i\,\textbf{Arg} (z)+2ni\,\pi \quad \quad(n\in \mathbb Z ) \end{eqnarray*}

Some basic properties of the function $\log z$ are the following:

  1. $\log \left(z_1 \,z_2\right)=\log z_1\log z_2$
  2. $\log\left( \frac{z_1}{z_2}\right)=\log z_1 -\log z_2$
  3. There may hold $\text{Log}\,\left(z_1 \,z_2\right)\neq \text{Log}\, z_1\text{Log}\, z_2$


Branches of Logarithms

From definition (\ref{log2}) let $\theta = \Theta + 2n\pi$ ($n\in \mathbb Z$), so we can write \begin{eqnarray}\label{log30} \log z = \ln r +i\theta. \end{eqnarray}

Now, let $\alpha$ be any real number. If we restrict the value of $\theta$ so that $\alpha < \theta < \alpha + 2n\pi$ , then the function \begin{eqnarray}\label{log3} \log z=\ln r +i\theta \quad (r> 0, \alpha < \theta < \alpha + 2\pi ), \end{eqnarray} with components \begin{eqnarray*} u(r, \theta)=\ln r, \quad v(r,\theta)=\theta, \end{eqnarray*} is a single-value and continuous function in the stated domain.

A branch of a multiple-valued function $f$ is any single-valued function $F$ that is analytic in some domain at each point $z$ of which the value $F(z)$ is one of the values of $f$. The requirement of analyticity, of course, prevents $F$ from taking on a random selection of the values of $f$.

Observe that for each fixed α, the single-valued function (\ref{log3}) is a branch of the multiple-valued function (\ref{log30}). The function \begin{eqnarray}\label{log4} \text{Log } z=\ln r +i\Theta \quad (r> 0, -\pi < \theta < \pi ), \end{eqnarray} is called the principal branch.

A branch cut is a portion of a line or curve that is introduced in order to define a branch $F$ of a multiple-valued function $f$. Points on the branch cut for $F$ are singular points of $F$, and any point that is common to all branch cuts of $f$ is called a branch point. The origin and the ray $\theta = \alpha$ make up the branch cut for the branch (\ref{log3}) of the logarithmic function. The branch cut for the principal branch (\ref{log4}) consists of the origin and the ray $= \pi$. The origin is evidently a branch point for branches of the multiple-valued logarithmic function.

We can visualize the multiple-valued nature of $\log z$ by using Riemann surfaces. The following interactive images show the real and imaginary components of $\log(z)$. Each branch of the imaginary part is identified with a different color.

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Special care must be taken in using branches of the logarithmic function, especially since expected identities involving logarithms do not always carry over from calculus.


Final Remark

Notice that for $z\neq 0$, we have \begin{eqnarray}\label{exp001} e^{\log z}=z\quad \text{and}\quad \log(e^z) =z+2n\pi i \end{eqnarray} with $n\in \mathbb Z .$

Example 3: Calculate $e^{\log z}$, and $\log \left(e^z\right)$ for $z=4i$.

Solution: If $z=4i$, then $e^z=e^{4i}$. Hence $$\log\left(e^{4i}\right)=4i +2n\pi i$$ with $n\in \mathbb Z .$ On the other hand, we have $$e^{\log\left(4i\right)}=4i.$$


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