18.1 Contour Integrals

Section 18.1: 2, 5, 7, 8, 11, 15, 19, 20, 21, 22, 24, 25, 26, 32, 33

$\int ^b _a f(z(t))z’(t)dt$ takes into account not just the function values along the curve, but also how fast the curve is traversed in the complex plane as $t$ varies from $a$ to $b$

Contour in the complex plane is a path that takes a complex variable $z$ takes through the plane.

1. Complex function is evaluated along the points of the curve by substituting $z(t)$ into $f(z)$
2. The differential $dz$ along the curve is computed by differentiating $z(t)$ with respect to $t$ to get $z’(t)dt$

Properties

The ML-inequality

If you have a continuous function $f(z)$ on a smooth curve $C$ and there exists a non-negative number $M$ for all $z$ on $C$, then the absolute value of the contour integral $f(z)$ along $C$ is:

$$ \left| \int_C f(z) \, dz \right| \leq M \cdot L $$

• $M$ is the maximum of the absolute value of $f(z)$ on $C$
• $L$ is the length of the contour $C$

• Reverse Triangle Inequality: $|a-b| ≥ ||a|-|b||$

Circulation and Net Flux

• Circulation: the tendency of a fluid to rotate around the curve

  $$ \text{circulation} = \text{Re} \left( \oint_C \bar {\phi(z)} \, dz \right) $$

  • The real part of the conjugate of$\oint _C f(z) dz$ gives circulation

• Net Flux: the rate at which fluid is flowing out of the contour

  $$ \text{net flux} = \text{Im} \left( \oint_C \bar {\phi(z)} \, dz \right) $$

  • Positive net flux means more fluid is exiting
  • Negative net flux means more fluid is entering
  • The imaginary part of the conjugate of $\oint _C f(z) dz$ gives circulation

18.2 Cauchy-Goursat Theorem

Section 18.2: 4, 5, 9, 10, 11, 12, 14, 17, 18, 20, 21, 22.

• Simply connected: if any simple closed curve within the domain can be continuously deformed or shrunk to a point without leaving the domain

  

  No holes or obstacles within the domain.

Cauchy’s Theorem: If a function $f(z)$ is analytic (differentiable) throughout a simply connected domain, then the integral of $f(z)$ around any closed contour $C$ in that domain is 0, or $\oint _c f(z) dz = 0$

Multiply Connected Domains

• Multiply connected: if there are one or more “holes” within the domain. These holes can be areas where the function is not defined or regions that are excluded from the domain