91Ó°ÊÓ

The given function is $f\left(x\right)=\{\begin{array}{llllll}-1,& -\mathrm{Ï€}& <& \mathrm{x}& <& \frac{\mathrm{Ï€}}{2}\\ 1,& \frac{\mathrm{Ï€}}{2}& <& \mathrm{x}& <& \mathrm{Ï€},\end{array}$.

The Fourier series for the function :

$$\begin{gathered} \mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{0}}}{\mathbf{2}}\mathbf{+}\underset{\mathbf{n}\mathbf{=}\mathbf{1}}{\overset{\mathbf{∞}}{\mathbf{∑}}}\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{cos}\mathbf{}\mathbf{nx}\mathbf{+}{\mathbf{b}}_{\mathbf{n}}\mathbf{sin}\mathbf{}\mathbf{nx}\mathbf{}\mathbf{)} \\ \mathbf{}\mathbf{}{\mathit{a}}_{\mathbf{0}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{π}}{\mathbf{∫}}_{\mathbf{-}\mathbf{π}}^{\mathbf{π}}\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathit{d}\mathit{x} \\ \mathbf{}\mathbf{}{\mathit{a}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{π}}{\mathbf{∫}}_{\mathbf{-}\mathbf{π}}^{\mathbf{π}}\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{n}\mathit{x}\mathit{d}\mathit{x} \\ \mathbf{}\mathbf{}{\mathit{b}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{π}}{\mathbf{∫}}_{\mathbf{-}\mathbf{π}}^{\mathbf{π}}\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{n}\mathit{x}\mathit{d}\mathit{x} \end{gathered}$$

If is an even function:

$$\begin{gathered} {\mathit{b}}_{\mathbf{n}}\mathbf{=}\mathbf{0} \\ \mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{0}}}{\mathbf{2}}\mathbf{+}\underset{\mathbf{n}\mathbf{=}\mathbf{1}}{\overset{\mathbf{∞}}{\mathbf{∑}}}{\mathbf{a}}_{\mathbf{n}}\mathbf{cos}\mathbf{}\mathbf{nx} \end{gathered}$$

If is an odd function:

$$\begin{gathered} {\mathit{a}}_{\mathbf{0}}\mathbf{=}{\mathit{a}}_{\mathbf{a}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0} \\ \mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\underset{\mathbf{n}\mathbf{=}\mathbf{1}}{\overset{\mathbf{∞}}{\mathbf{∑}}}{\mathit{b}}_{\mathbf{n}}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{nx} \end{gathered}$$

The sketch of the given function is shown below.

Coefficients of $a_0$:

$$\begin{gathered} a_0=\frac{1}{\mathrm{π}}{∫}_{-\mathrm{π}}^{\mathrm{π}}f\left(\mathrm{x}\right)dx \\ =\frac{1}{\mathrm{π}}\left\{-{∫}_{-\mathrm{π}}^{\frac{\mathrm{π}}{2}}dx+{∫}_{\frac{\mathrm{π}}{2}}^{\mathrm{π}}dx\right\} \\ =\frac{1}{\mathrm{π}}\left[{\left[x\right]}_{\mathrm{π}}^{\frac{\mathrm{π}}{2}}+{\left[x\right]}_{\frac{\mathrm{π}}{2}}^{\mathrm{π}}\right] \\ =-1 \end{gathered}$$

Coefficients of $a_n$:

$$\begin{gathered} a_{\mathrm{n}}=\frac{1}{\mathrm{Ï€}}{∫}_{-\mathrm{Ï€}}^{\mathrm{Ï€}}f\left(\mathrm{x}\right)\cos nxdx \\ =\frac{1}{\mathrm{Ï€}}\left[{∫}_{-\mathrm{Ï€}}^{\frac{\mathrm{Ï€}}{2}}\cos nxdx+{∫}_{\frac{\mathrm{Ï€}}{2}}^{\mathrm{Ï€}}\cos nxdx\right] \\ =\frac{1}{\mathrm{Ï€}}{\left[\sin nx\right]}_{\mathrm{Ï€}}^{\frac{\mathrm{Ï€}}{2}}+{\left[\sin nx\right]}_{\frac{\mathrm{Ï€}}{2}}^{\mathrm{Ï€}} \\ =\frac{2}{\mathrm{²ÔÏ€}}\sin \left(\frac{n\mathrm{Ï€}}{2}\right) \end{gathered}$$

Coefficients of $b_n$ are $\frac{1}{\mathrm{Ï€}},\frac{2}{2\mathrm{Ï€}},\frac{1}{3\mathrm{Ï€}},0,\frac{1}{5\mathrm{Ï€}},\frac{2}{6\mathrm{Ï€}},\frac{1}{7\mathrm{Ï€}}$.

Hence the expansion is $f\left(x\right)=\frac{1}{2}+\frac{2}{\mathrm{π}}\left\{\underset{n=1+4}{\overset{∞}{∑}}\frac{\cos nx}{n}-\underset{n=3+4}{\overset{∞}{∑}}\frac{\cos nx}{n}\right\}+\frac{2}{\mathrm{π}}\left\{-2\underset{n=2+4m}{\overset{∞}{∑}}\frac{\sin nx}{n}+2\underset{n=1+2m}{\overset{∞}{∑}}\frac{\sin nx}{n}\right\}$where $m=0,1,2,--.$.