Linear Combination of Contour Integrals

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Theorem

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

Let $f, g: \operatorname{Im} \left({C}\right) \to \C$ be continuous complex functions, where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$.

Let $z_0, z_1 \in \C$ be complex numbers.


Then:

$\displaystyle \int_C \left({ z_0 f \left({z}\right) + z_1 g \left({z}\right)} \right) \rd z = z_0 \int_C f \left({z}\right) \rd z + z_1 \int_C g \left({z}\right) \rd z$


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 $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

Then:

\(\displaystyle \int_C \left({ z_0 f \left({z}\right) + z_1 g \left({z}\right)} \right) \rd z\) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left({ z_0 f \left({\gamma_i \left({t}\right) }\right) + z_1 g \left({\gamma_i \left({t}\right) }\right)} \right) \gamma_i' \left({t}\right) \rd t\) Definition of Complex Contour Integral
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \left({ z_0 \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t + z_1 \int_{a_i}^{b_i} g \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t}\right)\) Linear Combination of Complex Integrals
\(\displaystyle \) \(=\) \(\displaystyle z_0 \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t + z_1 \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} g \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t\)
\(\displaystyle \) \(=\) \(\displaystyle z_0 \int_C f \left({z}\right) \rd z + z_1 \int_C g \left({z}\right) \rd z\)

$\blacksquare$


Sources