91Ó°ÊÓ

Expand each function \(f\) in a cosine series. Sketch the standardized extension of \(f\). $$ f(x)= \begin{cases}1 & \text { for } x \in I_{1}=\\{x: 0 \leqslant x<\pi / 2\\} \\ 0 & \text { for } x \in I_{2}=\\{x: \pi / 2 \leqslant x \leqslant \pi\\}\end{cases} $$

Find the Fourier series for the given function \(f\) $$ f(x)= \begin{cases}0 & \text { for } x \in I_{1}=\\{x:-\pi \leqslant x<0\\} \\\ 1 & \text { for } x \in I_{2}=\\{x ; 0 \leqslant x \leqslant \pi\\}\end{cases} $$

Find the Fourier expansion of \(f: x \rightarrow(1 / 3)\left(\pi^{2} x-x^{3}\right)\) on \(I=\\{x:-\pi \leqslant x \leqslant \pi\\}\). and show that \(\sum_{n=1}^{\infty} n^{-6}=\pi^{6} / 945\).

Find the Fourier series for the given function \(f\) $$ f(x)= \begin{cases}0 & \text { for } x \in I_{1}=\\{x:-\pi \leqslant x<\pi / 2\\} \\ 1 & \text { for } x \in I_{2}=\\{x: \pi / 2 \leqslant x \leqslant \pi\\}\end{cases} $$

Find the Fourier expansion for \(f\) given by $$ f: x \rightarrow\left\\{\begin{array}{rr} \frac{1}{2}\left(x^{2}-\pi x\right), & 0 \leqslant x \leqslant \pi \\ -\frac{1}{2}\left(x^{2}+\pi x\right), & -\pi \leqslant x \leqslant 0 \end{array}\right. $$

Find the Fourier series for the given function \(f\) $$ f(x)=x^{2} \quad \text { for } x \in I=\\{x:-\pi \leqslant x \leqslant \pi\\} $$

Find the Fourier series for the given function \(f\) $$ f(x)= \begin{cases}0 & \text { for } x \in I_{1}=\\{x:-\pi \leqslant x<0\\} \\\ x & \text { for } x \in I_{2}=\\{x: 0 \leqslant x \leqslant \pi\\} .\end{cases} $$

Expand each function \(f\) in a sine series. Sketch the standardized extension of \(f\). $$ f(x)=\left\\{\begin{aligned} 1 & \text { for } x \in I_{1}=\\{x: 0 \leqslant x<\pi / 2\\} \\ -1 & \text { for } x \in I_{2}=\\{x: \pi / 2 \leqslant x \leqslant \pi\\} \end{aligned}\right. $$

Find the Fourier series of the functions \(f\) and \(F\) given by $$ \begin{gathered} f: x \rightarrow|\sin x|, \quad-\pi \leqslant x \leqslant \pi, \\ F: x \rightarrow \begin{cases}-1+\cos x-\frac{2}{\pi} x & \text { for } I_{1}=\\{x:-\pi \leqslant x \leqslant 0\\} \\ 1-\cos x-\frac{2}{\pi} x & \text { for } I_{2}=\\{x: 0 \leqslant x \leqslant \pi\\} .\end{cases} \end{gathered} $$

Find the Fourier series for the given function \(f\) $$ f(x)=|\cos x| \quad \text { for } x \in I=\\{x:-\pi \leqslant x \leqslant \pi\\} $$