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.

$f\left(x\right)=\left\{\begin{array}{llllll}0,& -\pi & <& x& <& 0\\ 1,& 0& <& x& <& \frac{\pi }{2},\\ 0,& \frac{\pi }{2}& <& x& <& \pi .\end{array}\right.$

Short answer

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

Step-by-step solution

Given

The given function is $f\left(x\right)=\left\{\begin{array}{llllll}0,& -\pi & <& x& <& 0\\ 1,& 0& <& x& <& \frac{\pi }{2},\\ 0,& \frac{\pi }{2}& <& x& <& \pi .\end{array}\right.$

The concept of the Fourier series for the function 

The Fourier series for the function f (x):

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

If f(x)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 f(x) 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

Coefficients of :

$$\begin{gathered} a_0=\frac{1}{\mathrm{\pi }}{\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}f\left(\mathrm{x}\right)dx \\ =\frac{1}{\mathrm{\pi }}{\int }_{-\mathrm{\pi }}^{\frac{\mathrm{\pi }}{2}}dx \\ =\frac{1}{\mathrm{\pi }}{\left[x\right]}_0^{\frac{\mathrm{\pi }}{2}} \\ =\frac{1}{2} \end{gathered}$$

Calculate Coefficients of an

Coefficients of :

$$\begin{gathered} a_{\mathrm{n}}=\frac{1}{\mathrm{\pi }}{\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}f\left(\mathrm{x}\right)sinnxdx \\ =\frac{1}{\mathrm{\pi }}{\int }_0^{\frac{\mathrm{\pi }}{2}}f\left(\mathrm{x}\right)sindx \\ =\frac{1}{\mathrm{\pi }}{\left[\cos dx\right]}_0^{\frac{\mathrm{\pi }}{2}} \\ =\frac{1}{\mathrm{\pi }}\left[1-\cos \left(\frac{n\pi }{2}\right)\right] \end{gathered}$$

Calculate Coefficients of bn

Coefficients of ;

$\frac{1}{\pi },\frac{2}{2\pi },\frac{1}{3\pi },0,\frac{1}{5\pi },\frac{2}{6\pi },\frac{1}{7\pi }$

Hence the expansion is

$$\begin{gathered} \left[\sum _{n=1+4}^{\infty }{\displaystyle \frac{\cos nx}{n}}-\sum _{n=3+4}^{\infty }\frac{\cos nx}{n}-\sum _{n=1+2m}^{\infty }\frac{\sin nx}{n}+2\sum _{n=2+4m}^{\infty }\frac{\sin nx}{n}\right] \\ wherem=0,1,2,--. \end{gathered}$$