Contour

A curve in the complex plane is the set of points $z(t) = x(t) + iy(t)$, $a \le t \le b$, where $x(t)$ and $y(t)$ are continuous.

The curve is smooth iff:

• $z'(t)$ exists and is continuous on $[a,b]$
• $z'(t) \ne 0$ for every $t$ in $[a,b]$

A contour is a curve $C$ constructed by joining finitely many smooth curves end to end.

A smooth curve is a contour with 1 piece.

Properties

Orientation

The orientation of a contour is the direction in which its points are traced as $t$ increases.

The initial point is $z(a)$ and the terminal point is $z(b)$. Joining the smooth curves end to end fixes the direction along the whole contour.

Length

The length of a smooth curve given by $z(t)$, $a \le t \le b$, is

The length of a contour is the sum of the lengths of its smooth curves.

Types

Closed Contour

A contour is closed iff $z(a) = z(b)$.

Simple Contour

A contour is simple iff it does not cross itself, so $z(t_1) \ne z(t_2)$ for $t_1 \ne t_2$.

Simple Closed Contour

Also known as Jordan curve. A simple closed contour is a closed contour with $z(t_1) \ne z(t_2)$ for $a \le t_1 < t_2 < b$. Only the endpoints coincide.

The Jordan curve theorem states that the complement of a simple closed contour $C$ in the plane has exactly 2 connected components, and $C$ is the boundary of each.

• Interior
  The bounded component.
• Exterior
  The unbounded component.

Positive Orientation

A simple closed contour is positively oriented iff its interior lies on the left as it is traversed. Equivalently, it is traversed counterclockwise.

A negatively oriented contour is traversed clockwise.

Parametrization

A parametrization $z(t)$, $a \le t \le b$, must trace $C$ once in the required direction.

Line Segment

The segment from $z_1$ to $z_2$ is

Circle and Arcs

The circle of radius $r$ centered at $z_0$, traversed counterclockwise, is

An arc from angle $\theta_1$ to angle $\theta_2$ restricts the parameter range.

Examples:

• Upper semicircle of $|z| = r$ from $r$ to $-r$ uses $z(t) = re^{it}$, $0 \le t \le \pi$
• Lower semicircle of $|z| = r$ from $r$ to $-r$ uses $z(t) = re^{-it}$, $0 \le t \le \pi$
• Clockwise circle uses $z(t) = z_0 + re^{-it}$, $0 \le t \le 2\pi$

Ellipse

The ellipse with semi-axes $a$ and $b$ centered at $z_0$, traversed counterclockwise, is

Graph of a Function

A curve $y = g(x)$ from $x = a$ to $x = b$ uses $x$ as the parameter.

A curve $x = g(y)$ uses $y$ as the parameter.

Piecewise Contours

A polygon, such as a triangle or rectangle, is parametrized side by side. Each side is a line segment.

Reversed Orientation

Let $-C$ be the contour $C$ traversed in the opposite direction. It is parametrized by