Curves
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Definition (Curve): A curve $\gamma$ is a continuous complex valued function $\gamma(t)$ defined for $t$ in an interval of the real line.

Examples: The semi circle MATH for $t \in [0, \; \pi]$ is a curve in the complex plane. The straight line segment joining two distinct points $z_{1}$ and $z_{2}$ is given by MATH for $t \in [0, \; 1]$ is also an example for a curve in the complex plane.

Definition (Simple Curve): A curve $\gamma: \gamma(t)$ ( $a \le t \le b$) is said to be a simple curve if MATH for MATH. That is, simple curve is a curve that does not intersect itself (except for end points).

Examples: The circle MATH for $t \in [0, \; 2\pi]$ and the straight line segment MATH for $t \in [0, \; 1]$ are simple curves in the complex plane. Whereas, any curve of shape of the number $8$ is not a simple curve.
Definition (Closed Curve): A curve $\gamma: \gamma(t)$ ( $a \le t \le b$) is said to be a closed curve if MATH.

Examples: The circle MATH for $t \in [0, \; 2\pi]$ is a closed curve. The oriented boundary of a rectangle or a triangle are closed curves.

Definition (Simple Closed Curve or Jordan Curve): A curve $\gamma: \gamma(t)$ ( $a \le t \le b$) is said to be a simple closed curve or Jordan curve if MATH and MATH for MATH.

 
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