Line Integrals or Contour Integrals:
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The following properties of the line integral of complex valued functions can be easily proved.

  1. If $\alpha$ and $\beta$ are complex numbers and if $f(z)$ and $g(z)$ are (piecewise) continuous complex valued functions defined on a contour $C$, thenMATH
  2. Let $C$ be a contour consists of a contour $C_{1}$ from $z_{0}$ to $z_{1}$ followed by a contour $C_{2}$ from $z_{1}$ to $z_{2}$ where the initial point of $C_{2}$ is the final point of $C_{1}$. It is denoted by the notation $C = C_{1}+C_{2}$. If $f(z)$ is a (piecewise) continuous complex valued function on $C$, then MATH
  3. If $f(z)$ is a (piecewise) continuous complex valued function on a contour $C$ and if $-C$ is the opposite curve to $C$, then MATH
  4. f $f(z)$ is a (piecewise) continuous complex valued function on a contour MATH ( $a \le t \le b$), then MATH where $M$ is an upper bound for the set and $L$ is the length of the contour $C$.
 
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