Contour Integrals

A contour integral is essentially a line integral in the complex plane.

One trick we can use is that when the modulus of a function is constant, we can use the argument of $z$ as a parameter by writing $C:z=e^{\theta i},\arg \alpha \le \theta \le \arg \beta$

When the path of integration in a contour integral is a closed curve, we place a circle on the integral sign

$${\oint }_{C}f(z)dz.$$

You can add an arrow to the circle to indicate clockwise vs counterclockwise direction, but that's not supported in MathJax so I don't have it here!

Some additional properties:

$$\mathrm{Re}{\int }_{C}f(z)dz={\int }_C\mathrm{Re}[f(z)dz];$$

$$\mathrm{Im}{\int }_{C}f(z)dz={\int }_C\mathrm{Im}[f(z)dz];$$

$$|{\int }_{C}f(z)dz|\le {\int }_C|f(z)|\left|dz\right|\le ML$$

where $M$ is the maximum value of $|f(z)|$ for $z$ on $C$, and $L$ is the length of $C$.

Here's an example problem: evaluate ${\oint }_C{e}^{z}dz$ where $C$ is the closed (triangular) path from $0$ to $1$ ($C_1$) to $1+i$ ($C_2$) back to $0$ ($C_3$):

$${\oint }_C{e}^{z}dz={\int }_{C_1}{e}^{z}dz+{\int }_{C_2}{e}^{z}dz+{\int }_{C_3}{e}^{z}dz,$$

where

$$C_1:y=0,0\le x\le 1;$$

$$C_2:x=1,0\le y\le 1;$$

$$C_3:x=t,y=t,1\ge t\ge 0;$$

Thus,

$${\oint }_C{e}^{z}dz={\int }_0^1{e}^{x}dx+{\int }_0^1{e}^{1+yi}idy+{\int }_1^0{e}^{t+ti}(dt+dti)$$

$$={\left\{e^x\right\}}_0^1+{\left\{e^{1+yi}\right\}}_0^1+{\left\{e^{(1+i)t}\right\}}_1^0$$

$$=(e-1)+(e^{1+i}-e)+(1-e^{1+i})=0.$$

Some Important Contour Integrals

First, we'll compute ${\oint }_C\frac{1}{z}dz$ where $C$ is the unit circle. We'll call this out as a theorem, because it's a super important result that gets used all the time in complex analysis.