Sum of Complex Integrals on Adjacent Intervals

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

Let $f: \left[{a \,.\,.\, b}\right] \to \C$ be a continuous complex function.

Let $c \in \left[{a \,.\,.\, b}\right]$.


$\displaystyle \int_a^c f \left({t}\right) \ \mathrm dt + \int_c^b f \left({t}\right) \ \mathrm dt = \int_a^b f \left({t}\right) \ \mathrm dt$


From Continuous Complex Function is Complex Riemann Integrable, it follows that all three complex Riemann integrals are well defined.

From Real and Imaginary Part Projections are Continuous, it follows that $\operatorname{Re}: \C \to \R$ and $\operatorname{Im}: \C \to \R$ are continuous functions.

From Continuity of Composite Mapping, it follows that $\operatorname{Re} \circ f: \R \to \R$ and $\operatorname{Im} \circ f: \R \to \R$ are continuous real functions.


\(\displaystyle \int_a^b f \left({t}\right) \, \mathrm d t\) \(=\) \(\displaystyle \int_a^b \operatorname{Re} \left({f \left({t}\right) }\right) \, \mathrm d t + i \int_a^b \operatorname{Im} \left({f \left({t}\right) }\right) \, \mathrm d t\) Definition of Complex Riemann Integral
\(\displaystyle \) \(=\) \(\displaystyle \int_a^c \operatorname{Re} \left({f \left({t}\right) }\right) \, \mathrm d t + \int_c^b \operatorname{Re} \left({f \left({t}\right) }\right) \, \mathrm d t + i \left({\int_a^c \operatorname{Im} \left({f \left({t}\right) }\right) \, \mathrm d t + \int_c^b \operatorname{Im} \left({f \left({t}\right) }\right) \, \mathrm d t}\right)\) Sum of Integrals on Adjacent Intervals for Continuous Functions
\(\displaystyle \) \(=\) \(\displaystyle \int_a^c \operatorname{Re} \left({f \left({t}\right) }\right) \, \mathrm d t + i \int_a^c \operatorname{Im} \left({f \left({t}\right) }\right) \, \mathrm d t + \int_c^b \operatorname{Re} \left({f \left({t}\right) }\right) \, \mathrm d t + i \int_c^b \operatorname{Im} \left({f \left({t}\right) }\right) \, \mathrm d t\)
\(\displaystyle \) \(=\) \(\displaystyle \int_a^c f \left({t}\right) \, \mathrm d t + \int_c^b f \left({t}\right) \, \mathrm d t\)