Linear Combination of Contour Integrals

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Theorem

Let $C$ be a contour in $\C$.

Let $f, g: \Img C \to \C$ be continuous complex functions, where $\Img C$ denotes the image of $C$.

Let $\lambda, \mu \in \C$ be complex constants.


Then:

$\displaystyle \int_C \paren {\lambda \map f z + \mu \map g z} \rd z = \lambda \int_C \map f z \rd z + \mu \int_C \map g z \rd z$


Proof

By definition of contour, $C$ is a finite sequence $C_1, \ldots, C_n$ of directed smooth curves.

Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.

Then:

\(\ds \int_C \paren {\lambda \map f z + \mu \map g z} \rd z\) \(=\) \(\ds \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \paren {\lambda \map f {\map {\gamma_i} t} + \mu \map g {\map {\gamma_i} t} } \map {\gamma_i'} t \rd t\) Definition of Complex Contour Integral
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \paren {\lambda \int_{a_i}^{b_i} \map f {\map {\gamma_i} t} \map {\gamma_i'} t \rd t + \mu \int_{a_i}^{b_i} \map g {\map {\gamma_i} t} \map {\gamma_i'} t \rd t}\) Linear Combination of Complex Integrals
\(\ds \) \(=\) \(\ds \lambda \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \map f {\map {\gamma_i} t} \map {\gamma_i'} t \rd t + \mu \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \map g {\map {\gamma_i} t} \map {\gamma_i'} t \rd t\)
\(\ds \) \(=\) \(\ds \lambda \int_C \map f z \rd z + \mu \int_C \map g z \rd z\)

$\blacksquare$


Sources