Integrals of complex functions and Integration by parts
Trigonometry Complex Numbers Geometric Series 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 respected 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 $\omega$.

For example,

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

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

\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}


Evaluate $\displaystyle{\int_0^1 t e^{-j \omega t} dt}$, where $\omega$ is any complex number
We choose

\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}
\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}

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