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Integrals of Functions with Branch Cuts

When we consider multiple-valued functions, the path in a contour integral can contain a point on a branch cut of the integrand involved. The next two examples illustrate this.

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Example 1

Let $C$ be the semicircular path from $z_0=3$ to $z_1=-3$. That is $z(\theta) = 3 e^{i\theta}$, with $0\leq \theta \leq \pi$. Here we would like to evaluate the integral \begin{eqnarray}\label{ex-01} I = \int_C z^{1/2} dz. \end{eqnarray} To do so, we need to choose a particular branch of the multiple-valued function $z^{1/2}$. For example, we will use the principal branch \begin{eqnarray} |z|> 0, \; -\pi \lt \textbf{Arg}\,(z)\lt \pi. \end{eqnarray}

In this case, notice that although the principal branch of $z^{1/2}= \exp\left(\frac{1}{2}\text{Log}\, z\right)$ is not defined at the end point $z_1=-3$ of the contour $C$, the integral $I$ nevertheless exists.

Use the following applet to explore the value of $I$ for the given contour $C$. Just drag the points $z_0$ and $z_1$ to the corresponding values. You can also select other contours and explore what happens when they cross the branch cut $\{z\,:\, x =0 \text{ and } y \leq 0\}$.

As you already have figured it out, the integral (\ref{ex-01}) exists because the integrand is piecewise continuous on $C$. To confirm this, observe that when $z(\theta)= 3e^{i\theta}$, then $$ f\left(z(\theta)\right)= \exp\left[\frac{1}{2} \left(\ln 3 + i \theta\right)\right] = \sqrt{3}e^{i\theta/2}. $$

The left-hand limits of the real and imaginary components of the function \begin{eqnarray*} f\left(z(\theta)\right)z'(\theta) &=& \sqrt{3} e^{i\theta/2} \cdot 3ie^{i\theta } \\ &=& 3 \sqrt{3} i \,e^{i3\theta/2} \\ &=& -3\sqrt{3} \sin \frac{3\theta }{2} + i\, 3\sqrt{3} \cos \frac{3\theta }{2} \quad (0\leq \theta \lt \pi)\\ \end{eqnarray*} at $\theta = \pi $ exist. That is \begin{eqnarray*} \lim_{\theta \rightarrow \pi-}-3\sqrt{3} \sin \frac{3\theta }{2}= 3\sqrt{3} \quad \text{ and } \quad \lim_{\theta \rightarrow \pi-} 3\sqrt{3} \cos \frac{3\theta }{2} = 0.\\ \end{eqnarray*} This means that $f\left(z(\theta)\right)z'(\theta)$ is continuous on the closed interval $0 \leq \theta \leq \pi$ when its value at $\theta = \pi $ is defined as $3\sqrt{3}$. Therefore \begin{eqnarray*} I &=& \int_C f\left(z(\theta)\right)z'(\theta)\, d\theta= 3\sqrt{3} i \int_0^{\pi} e^{i\theta /2} d\theta = 3\sqrt{3} i\Bigg[ \bigg.\frac{2}{3i}e^{i3\theta /2}\bigg|_{0}^{\pi} \Bigg] \\ &=& 3\sqrt{3} i\Bigg[ -\frac{2}{3i}\left( 1+i\right)\Bigg]= -2\sqrt{3}(1+i). \end{eqnarray*}

Remark: Notice that $\int_C z^{1/2}\, dz=0$ for any circle not intersecting the branch cut $\{z\,:\, x =0 \text{ and } y \leq 0\}$. Why?

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Example 2

Consider the principal branch \begin{eqnarray}\label{br} f(z)=z^{i} = \exp \left[ i \,\text{Log } z \right]\; \text{ with }\; |z|>0,\, -\pi\lt \text{Arg } z \lt \pi \end{eqnarray} and $C$ the upper half circle from $z=-1$ to $z=1$; that is, $z(t)=-e^{-i \pi t}$ with $0 \leq t \leq 1$.

It is not difficult to verify that \begin{eqnarray}\label{ex-02} I=\int_C z^{i}\,dz &=& \frac{1+e^{-\pi}}{2}\,(1-i). \end{eqnarray} Use the following applet to confirm this. You can also analize $\int_C z^idz$ for other contours.

Remark 3: Notice that $\int_C z^{i}\, dz=0$ for any circle not intersecting the branch cut $\{z\,:\, x =0, y \leq 0\}$. Why?

In general, we can calculate $\int_C z^i dz$ for any contour $C$ from $z=-1$ to $z=1$ lying above the real axis. We just need to find an antiderivative of $z^i$. Unfortunately, we can not use the principal branch, defined in (\ref{br}), since this branch is not even defined at $z=-1$. But the integrand can be replaced by the branch $$z^{i} = \exp \left[ i \,\text{log } z \right]\; \text{ with }\; |z|>0,\, -\frac{\pi}{2}\lt \text{arg } z \lt \frac{3\pi}{2}.$$ since it agrees with the integrand along $C$.

Using an antiderivative of this new branch, we obtain \begin{eqnarray}\label{value} \int_{-1}^{1} z^i dz&=& \Bigg.\frac{z^{i+1}}{i+1}\Bigg|_{-1}^{1} = \frac{1+e^{-\pi}}{2}\,(1-i), \end{eqnarray} which is the same value as (\ref{ex-02}).

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