Linear Combination of Complex Integrals

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Theorem

Let $\closedint a b$ be a closed real interval.

Let $f, g: \closedint a b \to \C$ be complex Riemann integrable functions over $\closedint a b$.

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


Then:

$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$


Proof

First, we prove the result for addition only without multiplying by $\lambda, \mu$:

\(\ds \int_a^b \map f t + \map g t \rd t\) \(=\) \(\ds \int_a^b \map \Re {\map f t + \map g t} \rd t + i \int_a^b \map \Im {\map f t + \map g t} \rd t\) Definition of Complex Riemann Integral
\(\ds \) \(=\) \(\ds \int_a^b \map \Re {\map f t} + \map \Re {\map g t} \rd t\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds i \int_a^b \map \Im {\map f t} + \map \Im {\map g t} \rd t\) Addition of Real and Imaginary Parts
\(\ds \) \(=\) \(\ds \int_a^b \map \Re {\map f t} \rd t + \int_a^b \map \Re {\map g t} \rd t\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds i \paren {\int_a^b \map \Im {\map f t} \rd t + \int_a^b \map \Im {\map g t} \rd t}\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds \int_a^b \map f t \rd t + \int_a^b \map g t \rd t\)

$\Box$


Next, we show the result for only one complex integral multiplied by a constant $\lambda$.

By definition of complex number, we can find $\lambda_x, \lambda_y \in \R$ so $\lambda = \lambda_x + i \lambda_y$.

Then:

\(\ds \int_a^b \lambda \map f t \rd t\) \(=\) \(\ds \int_a^b \map \Re {\lambda \map f t} \rd t + i \int_a^b \map \Im {\lambda \map f t} \rd t\) Definition of Complex Riemann Integral
\(\ds \) \(=\) \(\ds \int_a^b \map \Re {\paren {\lambda_x + i \lambda_y} \map f t} \rd t\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds i \int_a^b \map \Im {\paren {\lambda_x + i \lambda_y} \map f t} \rd t\)
\(\ds \) \(=\) \(\ds \int_a^b \map \Re {\lambda_x \map f t} + \map \Re {i \lambda_y \map f t} \rd t\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds i \int_a^b \map \Im {\lambda_x \map f t} + \map \Im {i \lambda_y \map f t} \rd t\) Addition of Real and Imaginary Parts
\(\ds \) \(=\) \(\ds \int_a^b \lambda_x \map \Re {\map f t} - \lambda_y \map \Im {\map f t} \rd t\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds i \int_a^b \lambda_x \map \Im {\map f t} + \lambda_y \map \Re {\map f t} \rd t\) Multiplication of Real and Imaginary Parts
\(\ds \) \(=\) \(\ds \lambda_x \int_a^b \map \Re {\map f t} \rd t - \lambda_y \int_a^b \map \Im {\map f t} \rd t\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds i \lambda_x \int_a^b \map \Im {\map f t} \rd t + i \lambda_y \int_a^b \map \Re {\map f t} \rd t\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds \lambda_x \paren {\int_a^b \map \Re {\map f t} \rd t + i \int_a^b \map \Im {\map f t} \rd t}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds i \lambda_y \paren {i \int_a^b \map \Im {\map f t} \rd t + \int_a^b \map \Re {\map f t} \rd t}\)
\(\ds \) \(=\) \(\ds \lambda_x \int_a^b \map f t \rd t + i \lambda_y \int_a^b \map f t \rd t\) Definition of Complex Riemann Integral
\(\ds \) \(=\) \(\ds \lambda \int_a^b \map f t \rd t\)

$\Box$


It follows from the results above that:

\(\ds \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t\) \(=\) \(\ds \int_a^b \lambda \map f t \rd t + \int_a^b \mu \map g t \rd t\)
\(\ds \) \(=\) \(\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t\)

$\blacksquare$


Sources

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