4.4 Contours

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Recall that a path in $\mathbb{C}$ is any continuous function $\gamma :[a,b]\to \mathbb{C}$. We met paths in Week 2 when we discussed continuous functions. We will think of a path as describing a specific way of travelling from $\gamma (a)$ to $\gamma (b)$ in the complex plane. We will typically deal with simple paths like straight lines and circular arcs.

For any two complex numbers $z_1,z_2$ the straight-line from $z_1$ to $z_2$ can be parameterized by $\gamma :[0,1]\to \mathbb{C}$ with $\gamma (t)=(1-t)z_1+t{z}_2$.

Figure 1: The path $\gamma (t)=(1-t)z_1+t{z}_2$ on $[0,1]$ is a straight line from $z_1$ to $z_2$.

For any complex numbers $w$ and any $r>0$ the circular path centered at $w$ of radius $r$ can be parameterized by $\gamma :[0,2\pi ]\to \mathbb{C}$ with $\gamma (t)=w+r\mathsf{exp}(it)$.

Figure 2: The path $\gamma (t)=w+r\mathsf{exp}(it)$ on $[0,2\pi ]$ is a circular path around $w$ starting at $w+r$.

Definition

A path $\gamma :[a,b]\to \mathbb{C}$ is closed if $\gamma (b)=\gamma (a)$.

We will be using paths to define integrals. Not all paths are suited for this purpose. In this course we will work with smooth paths.

Definition

A smooth path is a path $\gamma$ with the property that its derivative $${\gamma }^{\prime }(t)=\underset{h\to 0}{lim}\frac{\gamma (t+h)-\gamma (t)}{t}$$ exists for every $t\in [a,b]$ and ${\gamma }^{\prime }$ is continuous on $[a,b]$.

Roughly speaking, a path is smooth if it has a continuously varying tangent line at every point. Thus smooth paths cannot have cusps or sharp corners.

Figure 3: A smooth path has a tangent line at every point $\gamma (t)$.

In a more advanced course we would work with the broader class of rectifiable curves. Rectifiability is a property that, amongst other things, guarantees that a path has a well-defined length. For smooth paths, we can define length by a formula.

Definition

The length of a smooth path $\gamma :[a,b]\to \mathbb{C}$ is $$\underset{a}{\overset{b}{\int }}|{\gamma }^{\prime }(t)|\mathsf{d}t$$ which is denoted $\ell (\gamma )$.

Note that $|{\gamma }^{\prime }(t)|$ is a real number so the integral defining the length does not involve complex analysis. It is a Riemann integral and can be calculated using techniques from Calculus.

Example

Calculate the lengths of the following smooth paths.

1. $\gamma (t)=(1-t)z_1+t{z}_2$ on $[0,1]$
2. $\gamma (t)=w+r{e}^{it}$ on $[0,2\pi ]$

Solution:

We calculate as follows.

1. ${\displaystyle \ell (\gamma )=\underset{0}{\overset{1}{\int }}|{\gamma }^{\prime }(t)|\mathsf{d}t=\underset{0}{\overset{1}{\int }}|-z_1+z_2|\mathsf{d}t=|z_2-z_1|}$
2. ${\displaystyle \ell (\gamma )=\underset{0}{\overset{2\pi }{\int }}|{\gamma }^{\prime }(t)|\mathsf{d}t=\underset{0}{\overset{2\pi }{\int }}|ir{e}^{it}|\mathsf{d}t=\underset{0}{\overset{2\pi }{\int }}r\mathsf{d}t=2\pi r}$

Often we will want to integrate over a number of paths joined together. One could make the latter a path by constructing a suitable reparametrisation, but in practice this makes things complicated; in particular the joins may not be smooth. It is simpler to give a name to several smooth paths joined together.

Definition

A contour is a finite list $({\gamma }_1,\dots ,{\gamma }_n)$ of smooth paths with the property that ${\gamma }_{i+1}$ starts where ${\gamma }_i$ ends for every $1\le i\le n-1$.

We usually write $\mathrm{\Gamma }=({\gamma }_1,\dots ,{\gamma }_n)$ for a contour.

Figure 4: A contour consists of a sequence paths, with successive paths beginning where preceding paths ended.

Definition

A contour $\mathrm{\Gamma }=({\gamma }_1,\dots ,{\gamma }_n)$ is closed if ${\gamma }_n$ ends where ${\gamma }_1$ begins.

If $\mathrm{\Gamma }=({\gamma }_1,\dots ,{\gamma }_n)$ is a contour then each of the ${\gamma }_i$ is a smooth path with a length $\ell ({\gamma }_i)$. It is then natural to define the length of a contour as follows.

Definition

The length of a contour $\mathrm{\Gamma }=({\gamma }_1,\dots ,{\gamma }_n)$ is the sum $$\ell (\mathrm{\Gamma })=\ell ({\gamma }_1)+\cdots +\ell ({\gamma }_n)$$ of the lengths of its constituent paths.

Example

Define ${\gamma }_1:[0,1]\to \mathbb{C}$ by ${\gamma }_1(t)=2t-1$ and ${\gamma }_2:[0,\pi ]\to \mathbb{C}$ by ${\gamma }_2(t)=e^{it}$. The contour $({\gamma }_1,{\gamma }_2)$ is a semicircle.

Its length is $$\underset{0}{\overset{1}{\int }}|{\gamma }_1^{\prime }(t)|\mathsf{d}t+\underset{0}{\overset{\pi }{\int }}|{\gamma }_2^{\prime }(t)|\mathsf{d}t=\underset{0}{\overset{1}{\int }}2\mathsf{d}t+\underset{0}{\overset{\pi }{\int }}|i{e}^{it}|\mathsf{d}t=2+\pi$$