Question

Sketch several periods of the corresponding periodic function of period $\mathbf{2}\mathit{\pi }$ . Expand the periodic function in a sine-cosine Fourier series.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{llllll}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{}& \mathbf{-}\mathbf{\pi }& \mathbf{<}& \mathbf{x}& \mathbf{<}& \frac{\mathbf{\pi }}{\mathbf{2}}\\ \mathbf{1}\mathbf{,}& \frac{\mathbf{\pi }}{\mathbf{2}}& \mathbf{<}& \mathbf{x}& \mathbf{<}& \mathit{\pi }\mathbf{,}\end{array}$

Short answer

The answer of the given function is:

$$\begin{gathered} f\left(x\right)=\frac{1}{4}+\frac{1}{\pi }\left\{\sum _{n=1+4}^{\infty }\frac{\cos nx}{n}-\sum _{n=3+4}^{\infty }\frac{\cos nx}{n}\right\}+\frac{2}{\pi }\left\{-2\sum _{n=2+4m}^{\infty }\frac{\sin nx}{n}+2\sum _{n=1+2m}^{\infty }\frac{\sin nx}{n}\right\} \\ wherem=0,1,2,--. \end{gathered}$$

Step-by-step solution

Given

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

The concept of the Fourier series for the function f(x)

The Fourier series for the function :

$$\begin{gathered} \mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{0}}}{\mathbf{2}}\mathbf{+}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{cos}\mathbf{}\mathbf{nx}\mathbf{+}{\mathbf{b}}_{\mathbf{n}}\mathbf{sin}\mathbf{}\mathbf{nx}\mathbf{}\mathbf{)} \\ \mathbf{}\mathbf{}{\mathit{a}}_{\mathbf{0}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\pi }}{\mathbf{\int }}_{\mathbf{-}\mathbf{\pi }}^{\mathbf{\pi }}\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathit{d}\mathit{x} \\ \mathbf{}\mathbf{}{\mathit{a}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\pi }}{\mathbf{\int }}_{\mathbf{-}\mathbf{\pi }}^{\mathbf{\pi }}\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{n}\mathit{x}\mathit{d}\mathit{x} \\ \mathbf{}\mathbf{}{\mathit{b}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\pi }}{\mathbf{\int }}_{\mathbf{-}\mathbf{\pi }}^{\mathbf{\pi }}\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{n}\mathit{x}\mathit{d}\mathit{x} \end{gathered}$$

If is an even function:

$$\begin{gathered} {\mathit{b}}_{\mathbf{n}}\mathbf{=}\mathbf{0} \\ \mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{0}}}{\mathbf{2}}\mathbf{+}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}{\mathbf{a}}_{\mathbf{n}}\mathbf{cos}\mathbf{}\mathbf{nx} \end{gathered}$$

If is an odd function:

$$\begin{gathered} {\mathit{a}}_{\mathbf{0}}\mathbf{=}{\mathit{a}}_{\mathbf{a}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0} \\ \mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}{\mathit{b}}_{\mathbf{n}}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{nx} \end{gathered}$$

From the given information

The sketch of the given function is shown below.

Coefficients of $a_0$:

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

Coefficients of $a_n$:

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

Calculate Coefficients of  bn

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

Hence the expansion is $f\left(x\right)=\frac{1}{2}+\frac{2}{\pi }\left\{\sum _{n=1+4}^{\infty }\frac{\cos nx}{n}-\sum _{n=3+4}^{\infty }\frac{\cos nx}{n}\right\}+\frac{2}{\pi }\left\{-2\sum _{n=2+4m}^{\infty }\frac{\sin nx}{n}+2\sum _{n=1+2m}^{\infty }\frac{\sin nx}{n}\right\}$where $m=0,1,2,--.$.