MAT 3730 Complex Variables Section 4.1 Contours

Preview Chapter 4: Complex Integration Very similar to line integrals in Multivariable Calculus 4.1: Set up the notations: Parametrizations Contours

Smooth Arcs

Smooth Closed Curves

Admissible Parametrizations

Example 1 (a) Find an admissible parametrization for the following smooth curve The straight-line segment from z 1 =-2-3i to z 2 =5+6i

Example 1 (b) Find an admissible parametrization for the following smooth curve The circle with radius 2 centered at 1-i

Example 1 (c) Find an admissible parametrization for the following smooth curve The graph of the function for

Directed Smooth Curves A smooth arc/closed curve is directed if its points have a specific ordering. (All curves in example 1 are directed with the order induced by the parametrization)

Contours

Opposite Contour

Definitions Closed contour The initial and terminal points coincide. Simple closed contour A closed contour with no multiple points other than its initial-terminal point. Example

Orientations A simple closed contour separates the plane into 2 domains: one bounded, and one unbounded. Positively oriented Negatively oriented

Length of a Smooth Curve

Next Class Read Section 4.2