# Linear Combination of Complex Integrals

## 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:

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

$\blacksquare$