Damped Exponential Cosine Integral

$\begin{displaymath} \int_0^\infty e^{-\omega T}\cos(\omega t)\,d\omega. \end{displaymath}$

(1)

Integrate by parts with

$\begin{displaymath} u\equiv e^{-\omega T}\qquad dv=\cos(\omega t)\,d\omega \end{displaymath}$

(2)

$\begin{displaymath} du\equiv -Te^{-\omega T}\,d\omega \qquad v={1\over t}\sin (\omega t), \end{displaymath}$

(3)

$\begin{displaymath} \int e^{-\omega T}\cos(\omega t)\,d\omega = {1\over t}e^{-\o... ...\omega t)+{T\over t}\int e^{-\omega T}\sin(\omega t)\,d\omega. \end{displaymath}$

Now integrate

$\begin{displaymath} \int e^{-\omega T}\sin(\omega t)\,d\omega \end{displaymath}$

(4)

by parts. Let

$\begin{displaymath} u=e^{-\omega T}\qquad dv=\sin(\omega t)\,d\omega \end{displaymath}$

(5)

$\begin{displaymath} du=-Te^{-\omega T}\,d\omega \qquad v=-{1\over t}\cos(\omega t), \end{displaymath}$

(6)

$\begin{displaymath} \int e^{-\omega t}\sin(\omega t)\,d\omega = -{1\over t}\cos(\omega t)-{T\over t} \int e^{-\omega T}\cos(\omega t)\,d\omega \end{displaymath}$

(7)

and

$\begin{displaymath} \int e^{\omega T} \cos(\omega t)\,d\omega = {1\over t}e^{-\o... ...ga t)-{T^2\over t^2} \int e^{-\omega T}\cos(\omega t)\,d\omega \end{displaymath}$

(8)

$\begin{displaymath} \left({1+{T^2\over t^2}}\right)\int e^{-\omega T}\cos(\omega... ...{{1\over t} \sin(\omega t)-{T\over t^2} \cos(\omega t)}\right] \end{displaymath}$

(9)

$\begin{displaymath} {t^2+T^2\over t^2} \int e^{-\omega T}\cos(\omega t)\,d\omega = {e^{-\omega t}\over t^2} [t\sin(\omega T) - T\cos(\omega t)] \end{displaymath}$

(10)

Therefore,

$\begin{displaymath} \int_0^\infty e^{-\omega T}\cos(\omega t)\,d\omega= 0+{T\over t^2+T^2}={T\over t^2+T^2}. \end{displaymath}$

(11)