Complex line integrals

Section 1.13 Complex line integrals

Convention 1.13.1. Paths and their images.

From now on we reserve the right to conflate a path \(\gamma\colon [a,b]\rightarrow \C\) with its image

\begin{equation*} \im \phi=\{\phi(t)\colon t\in [a,b]\}\text{.} \end{equation*}

We will take care not to abuse this privilege.

Definition 1.13.2. Complex line integral.

Let \(\gamma\colon [a,b]\rightarrow \C\) be smooth path. Given a complex function \(f\) defined on \(\gamma\text{,}\) the integral of \(f\) over \(\gamma\), denoted \(\int\limits_{\gamma}f\, dz\text{,}\) is defined as

\begin{equation} \int\limits_{\gamma}f\, dz=\int_a^bf(\gamma(t))\, \gamma'(t)\, dt\text{,}\tag{1.49} \end{equation}

provided this integral exists.

If \(\gamma\) is the concatenation of smooth curves \(\gamma_k\text{,}\) \(1\leq k\leq n\text{,}\) then we define

\begin{equation} \int_\gamma f\, dz=\sum_{k=1}^n\int_{\gamma_k}f\, dz\text{.}\tag{1.50} \end{equation}

Integrals as in (1.49) or (1.50) are called complex line integrals (or complex contour integrals).

Remark 1.13.3. Connection with Riemann integral.

It can be shown that the complex line integral \(\int\limits_\gamma f\, dz\) for a piecewise smooth curve \(\gamma\colon[a,b]\rightarrow \C\) has an equivalent definition as the limit of Riemann-like sums of the form

\begin{equation*} \sum_{k=0}^{n-1} f(z_k^*)(z_{k+1}-z_k)\text{,} \end{equation*}

where the points \(z_0,z_1,\dots, z_n\) partition the curve \(\mathcal{C}=\{\gamma(t)\colon t\in [a,b]\}\) into curve segments whose arc lengths approach zero as \(n\to \infty\text{.}\) In this sense the complex line integral over a general smoothly parametrized curve in \(\C\) can be seen directly as an extension of the Riemann integral of function along a line segment \([a,b]\text{.}\)

Example 1.13.4. Classic line integral.

Fix \(R> 0\text{.}\) Show that

\begin{equation*} \int_{\gamma}\frac{1}{z}\, dz=2\pi i\text{,} \end{equation*}

where \(\gamma(t)=Re^{it}\text{,}\) \(t\in [0,2\pi]\text{.}\)

More generally, show that

\begin{equation} \int_{\gamma}\frac{1}{z-z_0}\, dz=2\pi i\tag{1.51} \end{equation}

for any fixed \(z_0\in C\text{,}\) where \(\gamma(t)=z_0+Re^{it}\text{,}\) \(t\in [0,2\pi]\text{.}\)

Solution.

Fix a piecewise smooth complex path \(\gamma\text{.}\) Since the line integral \(\int_\gamma f\, dz\) is defined as the complex-valued function integral \(\int_a^bf(\gamma(t))\gamma'(t)\, dt\text{,}\) all the usual properties of integration also hold for complex line integrals. For example:

\(\int_\gamma cf+dg\, dz=c\int_\gamma f\, dz+d\int_\gamma g\, dz\) for any \(\gamma\)-integrable functions \(f, g\) and complex constants \(c,d\in \C\text{.}\)
\(\int_{-\gamma}f\, dz=-\int_{\gamma}f\, dz\) for any \(\gamma\)-integrable function \(f\text{.}\)
If \(\gamma'\) is a reparametrization of \(\gamma\text{,}\) then \(\int_\gamma f\, dz=\int_{\gamma'}f\, dz\) for all \(\gamma\)-integrable functions \(f\text{.}\)

We also have a fundamental theorem of calculus for line integrals.

Theorem 1.13.5. Line integrals: fundamental theorem of calculus.

Let \(\gamma\colon [a,b]\rightarrow \C\) be piecewise smooth, let \(U\) be an open set containing \(\im \phi=\{\gamma(t)\colon t\in [a,b]\}\text{,}\) and suppose \(f\colon U\rightarrow \C\) is continuous. If \(F\colon U\rightarrow \C\) satisfies \(F'(z)=f(z)\) for all \(z\in U\text{,}\) then

\begin{equation} \int_\gamma f\, dz=F(\gamma(b))-F(\gamma(a))=F(z_1)-F(z_0)\text{,}\tag{1.52} \end{equation}

where \(z_0=\gamma(a)\) and \(z_1=\gamma(b)\) are the initial and terminal points of \(\gamma\text{,}\) respectively.

Proof.

Example 1.13.6. Line integral: FTC.

Let \(\gamma\colon [a,b]\rightarrow \C\) be a piecewise smooth closed curve. Prove: \(\int_\gamma \sin(iz)\, dz=0\text{.}\)

Solution.

Definition 1.13.7. Antiderivative.

Let \(f\) be a complex function defined on the open set \(U\text{.}\) An antiderivative of \(f\) on \(U\) is a complex function \(F\) satisfying \(F'(z)=f(z)\) for all \(z\in U\text{.}\)

Corollary 1.13.8.

Let \(f\) be a complex function that is continuous on the open set \(U\text{.}\) If \(f\) has an antiderivative on \(U\text{,}\) then

\begin{equation*} \int_\gamma f\, dz=0 \end{equation*}

for any closed piecewise smooth curve \(\gamma\) with image lying in \(U\text{.}\)

Theorem 1.13.9. ML-inequality.

Let \(\gamma\colon [a,b]\rightarrow \C\) be piecewise smooth, and let \(L\) be the arclength of \(\gamma\text{.}\) If \(f\) is \(\gamma\)-integrable and satisfies \(\abs{f(\gamma(t))}\leq M\) for all \(t\in [a,b]\text{,}\) then

\begin{equation} \abs{\int_\gamma f\, dz}\leq M\, L\text{.}\tag{1.53} \end{equation}

Proof.

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