Integrals of Complex Functions And Integration By Parts

In this class, we will deal only with integrals of complex functions of a real variable integrated with respect to the real variable. This is identical to integration of real functions of real variables. The $j$ is simply treated as a constant in all these cases. In this class, we are typically interested in time $t$ or frequency $\omega$ being the independent variable. Hence, the integrals will be with respect to $t$ or $\omega$.

Example: $\int_0^{\pi/4} e^{j 2 t} \ dt = \left[\frac{e^{j 2 t}}{2 j}\right]_{0}^{\pi/4} = \frac{j-1}{2j}$

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Sometimes we will have to use integration by parts to evaluate integrals. The main result to recall is

(1)

\begin{align} \displaystyle{\int_a^b u(t) \ dv(t) \ dt = \left[ u(t) \ v(t) \right]_a^b - \int_a^b v(t) \ du(t) \ dt} \end{align}

Example: If we want to evaluate $\displaystyle{\int_0^1 t e^{-j \omega t} dt}$, where $\omega$ is any complex number
We choose

(2)

\begin{eqnarray} \nonumber u(t) = t &,& dv(t) = e^{-j \omega t} \ dt \\ \nonumber \Longrightarrow du(t)=dt &,& v(t) = \frac{e^{-j \omega t}}{-j \omega} \end{eqnarray}

(3)

\begin{eqnarray} \nonumber \int_0^1 t e^{-j \omega t} dt & = & \displaystyle{\left[\frac{t e^{-j \omega t}}{-j \omega}\right]_0^1 - \int_0^1 \frac{e^{-j \omega t}}{-j \omega} \ dt}\\ \nonumber & = & - e^{-j \omega} - \displaystyle{\left[\frac{e^{-j \omega t}}{-\omega^2} \right]_0^1} = - e^{-j \omega} + \frac{e^{-j \omega}-1}{\omega^2} \end{eqnarray}