Question

Repeat the example using the same Fourier series but at $\mathit{x}\mathbf{=}\mathit{\pi }\mathbf{/}\mathbf{2}$.

Short answer

At, $x=\frac{\mathrm{\pi }}{2}$:

$$\begin{gathered} f\left(\frac{1}{2}\right)=\left(\begin{array}{c}\frac{1}{4}+\frac{1}{\mathrm{\pi }}\left(\frac{1}{1}\cos 3x+\frac{1}{5}\cos 5x-\frac{1}{7}\cos 7x+...\right)\\ +\frac{1}{\mathrm{\pi }}\left(\frac{1}{1}\sin x-\frac{2}{2}\sin 2x+\frac{1}{3}\sin 3x+\frac{1}{5}\sin 5x+\frac{2}{6}\sin 6x+\frac{1}{7}\sin 7x+...\right)\end{array}\right) \\ \frac{\mathrm{\pi }}{4}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+... \end{gathered}$$

Step-by-step solution

Given

The given function is $f\left(x\right)=\left\{\begin{array}{l}0,-\mathrm{\pi }<\mathrm{x}<0\\ \sin x,0<x<\mathrm{\pi }\end{array}\right.$

The given point is $x=\frac{\mathrm{\pi }}{2}$.

Definition of Fourier series

The Fourier series for the function $\mathit{f}\left(x\right)$:

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

If $\mathit{f}\left(x\right)$is an even function:

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

If $\mathit{f}\left(x\right)$is an odd function:

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

Write the Fourier series of the function and input the value

The Fourier series of function is shown below.

$$\begin{gathered} f\left(x\right)-\left(\begin{array}{c}\frac{1}{4}+\frac{1}{\mathrm{\pi }}\left(\frac{1}{1}\cos 3x+\frac{1}{5}\cos 5x-\frac{1}{7}\cos 7x+...\right)\\ +\frac{1}{\mathrm{\pi }}\left(\frac{1}{1}\sin x-\frac{2}{2}\sin 2x+\frac{1}{3}\sin 3x+\frac{1}{5}\sin 5x+\frac{2}{6}\sin 6x+\frac{1}{7}\sin 7x+...\right)\end{array}\right) \\ \frac{\mathrm{\pi }}{4}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+... \end{gathered}$$

Now, at $x-\frac{\mathrm{\pi }}{2}$.

role="math" localid="1659238212014" $$\begin{gathered} f\left(\frac{1}{2}\right)-\left(\begin{array}{c}\frac{1}{4}+\frac{1}{\mathrm{\pi }}\left(\frac{1}{1}\cos 3x+\frac{1}{5}\cos 5x-\frac{1}{7}\cos 7x+...\right)\\ +\frac{1}{\mathrm{\pi }}\left(\frac{1}{1}\sin x-\frac{2}{2}\sin 2x+\frac{1}{3}\sin 3x+\frac{1}{5}\sin 5x+\frac{2}{6}\sin 6x+\frac{1}{7}\sin 7x+...\right)\end{array}\right) \\ \frac{\mathrm{\pi }}{4}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+... \end{gathered}$$