So far we have always supposed that the subject of integration in a definite integral is real. We define the integral of a complex function $$f(x) = {\phi}(x) + i\psi(x)$$ of the real variable $$x$$, between the limits $$a$$ and $$b$$, by the equations $\int_{a}^{b} f(x)\, dx = \int_{a}^{b} \{\phi(x) + i\psi(x)\}\, dx = \int_{a}^{b} \phi(x)\, dx + i \int_{a}^{b} \psi(x)\, dx;$ and it is evident that the properties of such integrals may be deduced from those of the real integrals already considered.

There is one of these properties that we shall make use of later on. It is expressed by the inequality $\begin{equation*} \left|\int_{a}^{b} f(x)\, dx\right| \leq \int_{a}^{b} |f(x)|\, dx \tag{1} \end{equation*}$ This inequality may be deduced without difficulty from the definitions of §§156 and 157. If $$\delta_{\nu}$$ has the same meaning as in § 156, $$\phi_{\nu}$$ and $$\psi_{\nu}$$ are the values of $$\phi$$ and $$\psi$$ at a point of $$\delta_{\nu}$$, and $$f_{\nu} = \phi_{\nu} + i\psi_{\nu}$$, then we have \begin{aligned} \int_{a}^{b} f\, dx = \int_{a}^{b} \phi\, dx + i \int_{a}^{b} \psi\, dx &= \lim \sum \phi_{\nu}\, \delta_{\nu} + i \lim \sum \psi_{\nu}\, \delta_{\nu} \\ &= \lim \sum (\phi_{\nu} + i\psi_{\nu})\, \delta_{\nu} = \lim \sum f_{\nu}\, \delta_{\nu},\end{aligned} and so $\int_{a}^{b} f\, dx = |\lim \sum f_{\nu}\, \delta_{\nu}| = \lim |\sum f_{\nu}\, \delta_{\nu}|;$ while $\int_{a}^{b} |f|\, dx = \lim \sum |f_{\nu}|\, \delta_{\nu}.$ The result now follows at once from the inequality $|\sum f_{\nu}\, \delta_{\nu}| \leq \sum |f_{\nu}|\, \delta_{\nu}.$

It is evident that the formulae (1) and (2) of § 162 remain true when $$f$$ is a complex function $$\phi + i\psi$$.