Subsection 5.3.1 Summing of Fourier Series
Since the Fouries series is a convergent series, it converges to
- \(f(x)\text{,}\) if \(x\) is a point of continuity.
- \(\frac{f(x+0) + f(x-0)}{2}\text{,}\) if \(x\) is a point of discontinuity.
- \(\frac{f(-\pi+0) + f(\pi-0)}{2}\) at \(x=\pm \pi\text{,}\) if \(x\) is a point of discontinuity.
Subsubsection 5.3.1.1 Half - Range Series
If a half - range series for a function \(f(x)\) is desired, then the function is generally defined in an open interval \([0.\pi]\text{,}\) which is half of the interval \([-\pi, \pi]\) and hence the name half - range. In this case the function \(f(x)\) is expressed as an odd or even function. For interval (0,c)
\begin{equation*}
a_{0}= \frac{2}{c}\int\limits_{0}^{c}f(x)\,dx;
\end{equation*}
\begin{equation*}
a_{n}= \frac{2}{c}\int\limits_{0}^{c} f(x) \cos (\frac{n\pi x}{c}) \,dx;
\end{equation*}
\begin{equation*}
b_{n}= \frac{2}{c}\int\limits_{0}^{c} f(x) \sin (\frac{n\pi x}{c}) \,dx
\end{equation*}
