Subsection 5.5.2 Fourier Integral
Fourier series of a periodic function \(f(x)\) in the interval [-l,l] is given by
\begin{equation}
f(x) = \frac{a_{o}}{2} + \sum\limits_{n=1}^{\infty}\left[a_{n}\cos \left(\frac{n\pi x}{l}\right) + b_{n}\sin \left(\frac{n\pi x}{l}\right)\right] \tag{5.5.4}
\end{equation}
where,
\begin{equation*}
a_{0}=\frac{1}{l}\int\limits_{-l}^{l}f(t) \,dt,
\end{equation*}
\begin{equation*}
a_{n} = \frac{1}{l}\int\limits_{-l}^{l}f(t) \cos \left(\frac{n\pi t}{l}\right)\,dt,
\end{equation*}
and
\begin{equation*}
b_{n} = \frac{1}{l}\int\limits_{-l}^{l}f(t) \sin \left(\frac{n\pi t}{l}\right)\,dt.
\end{equation*}
Substituting these values in equation (5.5.4), we get -
\begin{equation*}
f(x) = \frac{1}{2l}\int\limits_{-l}^{l}f(t) \,dt
\end{equation*}
\begin{equation*}
+ \frac{1}{l}\sum\limits_{n=1}^{\infty}\left[\int\limits_{-l}^{l}f(t)\left\{\cos \left(\frac{n\pi x}{l}\right) \cos \left(\frac{n\pi t}{l}\right)\right.
\end{equation*}
\begin{equation*}
\left.+ \sin \left(\frac{n\pi x}{l}\right)\sin \left(\frac{n\pi t}{l}\right)\right\}\,dt \right]
\end{equation*}
or,
\begin{equation}
f(x) = \frac{1}{2l}\int\limits_{-l}^{l}f(t) \,dt+ \frac{1}{l}\sum\limits_{n=1}^{\infty}f(t) \cos \left[\frac{n\pi}{l}(x-t)\right]\,dt\tag{5.5.5}
\end{equation}
If we set \(u=\frac{n\pi}{l} \) so that \(\bigtriangleup u= \frac{\bigtriangleup n\pi}{l}, \) where \(\bigtriangleup n = 1 \) and \(\frac{1}{l}=\frac{\bigtriangleup u} {\pi}.\) Hence, equation (5.5.5) becomes
\begin{equation*}
f(x) = \frac{1}{2l}\int\limits_{-l}^{l}f(t) \,dt+ \frac{1}{\pi}\sum\limits_{u=\pi/l}^{\infty}\bigtriangleup u \int\limits_{-l}^{l}f(t) \cos \left[u(x-t)\right]\,dt
\end{equation*}
or,
\begin{equation}
f(x) = \frac{1}{\pi}\int\limits_{0}^{\infty}\, du \int\limits_{-\infty}^{\infty}[u(x-t)]\,dt \tag{5.5.6}
\end{equation}
Since
\begin{equation*}
\frac{1}{2l}\int\limits_{-l}^{l}f(t) \,dt \rightarrow 0,
\end{equation*}
and
\begin{equation*}
\sum\limits_{u=\pi/l}^{\infty}\bigtriangleup u \rightarrow \int\limits_{0}^{\infty} \,du
\end{equation*}
[as \(l\rightarrow \infty\)]. Equation (5.5.6) is called the Fourier integral formula.
