Linear Combination of Complex Integrals

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Theorem

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f, g: \left[{a \,.\,.\, b}\right] \to \C$ be complex Riemann integrable over $\left[{a \,.\,.\, b}\right]$.

Let $w, z \in \C$ be complex numbers.


Then:

$\displaystyle \int_a^b \left({w f \left({t}\right) + z g \left({t}\right) }\right) \ \mathrm dt = w \int_a^b f \left({t}\right) \ \mathrm dt + z \int_a^b g \left({t}\right) \ \mathrm dt$


Proof

First, we prove the result for addition only without multiplying by $w, z$:

\(\displaystyle \int_a^b f \left({t}\right) + g \left({t}\right) \ \mathrm dt\) \(=\) \(\displaystyle \int_a^b \operatorname{Re} \left({f \left({t}\right) + g \left({t}\right) }\right) \ \mathrm dt + i \int_a^b \operatorname{Im} \left({f \left({t}\right) + g \left({t}\right) }\right) \ \mathrm dt\) by definition of complex Riemann integral
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \operatorname{Re} \left({f \left({t}\right) }\right) + \operatorname{Re} \left({ g \left({t}\right) }\right) \ \mathrm dt\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle i \int_a^b \operatorname{Im} \left({f \left({t}\right) }\right) + \operatorname{Im} \left({g \left({t}\right) }\right) \ \mathrm dt\) by Addition of Real and Imaginary Parts
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \operatorname{Re} \left({f \left({t}\right) }\right) \ \mathrm dt + \int_a^b \operatorname{Re} \left({ g \left({t}\right) }\right) \ \mathrm dt\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle i \left({ \int_a^b \operatorname{Im} \left({f \left({t}\right) }\right) \ \mathrm dt + \int_a^b \operatorname{Im} \left({g \left({t}\right) }\right) \ \mathrm dt}\right)\) by Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b f \left({t}\right) \ \mathrm dt + \int_a^b g \left({t}\right) \ \mathrm dt\)

$\Box$


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

By definition of complex number, we can find $x, y \in \R$ so $z = x + iy$.

Then:

\(\displaystyle \int_a^b z f \left({t}\right) \ \mathrm dt\) \(=\) \(\displaystyle \int_a^b \operatorname{Re} \left({z f \left({t}\right) }\right) \ \mathrm dt + i \int_a^b \operatorname{Im} \left({z f \left({t}\right) }\right) \ \mathrm dt\) by definition of complex Riemann integral
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \operatorname{Re} \left({\left({x + iy} \right) f \left({t}\right) }\right) \ \mathrm dt\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle i \int_a^b \operatorname{Im} \left({\left({x + iy} \right) f \left({t}\right) }\right) \ \mathrm dt\)
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \operatorname{Re} \left({x f \left({t}\right) }\right) + \operatorname{Re} \left({i y f \left({t}\right) }\right) \ \mathrm dt\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle i \int_a^b \operatorname{Im} \left({x f \left({t}\right) }\right) + \operatorname{Im} \left({i y f \left({t}\right) }\right) \ \mathrm dt\) by Addition of Real and Imaginary Parts
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b x \operatorname{Re} \left({f \left({t}\right) }\right) - y \operatorname{Im} \left({f \left({t}\right) }\right) \ \mathrm dt\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle i \int_a^b x \operatorname{Im} \left({f \left({t}\right) }\right) + y \operatorname{Re} \left({f \left({t}\right) }\right) \ \mathrm dt\) by Multiplication of Real and Imaginary Parts
\(\displaystyle \) \(=\) \(\displaystyle x \int_a^b \operatorname{Re} \left({f \left({t}\right) }\right) \ \mathrm dt - y \int_a^b \operatorname{Im} \left({f \left({t}\right) }\right) \ \mathrm dt\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle i x \int_a^b \operatorname{Im} \left({f \left({t}\right) }\right) \ \mathrm dt + i y \int_a^b \operatorname{Re} \left({f \left({t}\right) }\right) \ \mathrm dt\) by Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle x \left({\int_a^b \operatorname{Re} \left({f \left({t}\right) }\right) \ \mathrm dt + i \int_a^b \operatorname{Im} \left({f \left({t}\right) }\right) \ \mathrm dt }\right)\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle i y\left({i \int_a^b \operatorname{Im} \left({f \left({t}\right) }\right) \ \mathrm dt + \int_a^b \operatorname{Re} \left({f \left({t}\right) }\right) \ \mathrm dt }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle x\int_a^b f \left({t}\right) \ \mathrm dt + i y\int_a^b f \left({t}\right) \ \mathrm dt\) by definition of complex Riemann integral
\(\displaystyle \) \(=\) \(\displaystyle z \int_a^b f \left({t}\right) \ \mathrm dt\)

$\Box$


It follows from the results above that:

\(\displaystyle w \int_a^b f \left({t}\right) \ \mathrm dt + z \int_a^b g \left({t}\right) \ \mathrm dt\) \(=\) \(\displaystyle \int_a^b wf \left({t}\right) \ \mathrm dt + \int_a^b zg \left({t}\right) \ \mathrm dt\)
\(\displaystyle \) \(=\) \(\displaystyle \int_a^b \left({w f \left({t}\right) + z g \left({t}\right) }\right) \ \mathrm dt\)

$\blacksquare$


Sources