\[f\left({x}\right)=x^{2},\quad \]

Функция удовлетворяет условиях Дирихле и может быть разложена в ряд Фурье.

Ряд Фурье общего вида для \(2\pi\) периодичных функций:

\[f\left( x \right) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {\left( {{a_{n}}\cos nx + {b_{n}}\sin nx} \right)}\]

Вычислим коэффициенты ряда:

\[a_0=\frac{1}{\pi}\int\limits_{0}^{\pi}{f\left({x}\right)dx}=\int\limits_{- \pi}^{\pi} {\frac{1}{\pi}\cdot x^{2}dx} =\left.{\frac{x^{3}}{3 \pi}}\right | _{- \pi}^{\pi} =\frac{\pi^{3}}{3 \pi}- \frac{\left(- \pi\right)^{3}}{3 \pi}=\frac{2 \pi^{2}}{3} \approx 6.5797\]

\[a_{n} = \frac{1}{L}\int\limits_{-\pi}^{\pi} {f\left( x \right)\cos nxdx} =\frac{1}{\pi}\int\limits_{- \pi}^{\pi} {x^{2}\cos{\left(n x \right)}dx}=\left| {\begin{array}{*{20}{c}}u=x^{2}&du=2 xdx\\v=-\frac{1}{\pi n}\sin{\left(n x \right)}&dv=\frac{1}{\pi}\cos{\left(n x \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}x^{2} \sin{\left(n x \right)}} \right|_{- \pi}^{\pi}+\frac{1}{\pi n}\int\limits_{- \pi}^{\pi} {2 x \sin{\left(n x \right)}dx}=- \frac{2 \pi \sin{\left(\pi n \right)}}{n}+\frac{1}{\pi n}\int\limits_{- \pi}^{\pi} {2 x \sin{\left(n x \right)}dx}=\left| {\begin{array}{*{20}{c}}u=2 x&du=2dx\\v=\frac{1}{\pi n^{2}}\cos{\left(n x \right)}&dv=\frac{1}{\pi n}\sin{\left(n x \right)}dx\end{array}} \right| = - \frac{2 \pi \sin{\left(\pi n \right)}}{n}\left. {+\frac{1}{\pi n^{2}}2 x \cos{\left(n x \right)}} \right|_{- \pi}^{\pi}-\frac{1}{\pi n^{2}}\int\limits_{- \pi}^{\pi} {2 \cos{\left(n x \right)}dx}=\frac{- 2 \pi^{2} \sin{\left(\pi n \right)} + 4 \pi \cos{\left(\pi n \right)}}{\pi n^{3}}+\frac{2}{\pi n^{3}}\left. {\sin{\left(n x \right)}} \right|_{- \pi}^{\pi}=\frac{- 2 \pi^{2} \sin{\left(\pi n \right)} + 4 \pi \cos{\left(\pi n \right)}}{\pi n^{2}} + \frac{4 \sin{\left(\pi n \right)}}{\pi n^{3}}=\frac{4 \left(-1\right)^{n}}{n^{2}}\]

\[b_{n} = \frac{1}{L}\int\limits_{-\pi}^{\pi} {f\left( x \right)\sin nxdx} =\frac{1}{\pi}\int\limits_{- \pi}^{\pi} {x^{2}\sin{\left(n x \right)}dx}=\left| {\begin{array}{*{20}{c}}u=x^{2}&du=2 xdx\\v=-\frac{1}{\pi n}\cos{\left(n x \right)}&dv=\frac{1}{\pi}\sin{\left(n x \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}x^{2} \cos{\left(n x \right)}} \right|_{- \pi}^{\pi}+\frac{1}{\pi n}\int\limits_{- \pi}^{\pi} {2 x \cos{\left(n x \right)}dx}=+\frac{1}{\pi n}\int\limits_{- \pi}^{\pi} {2 x \cos{\left(n x \right)}dx}=\left| {\begin{array}{*{20}{c}}u=2 x&du=2dx\\v=\frac{1}{\pi n^{2}}\sin{\left(n x \right)}&dv=\frac{1}{\pi n}\cos{\left(n x \right)}dx\end{array}} \right| = \left. {\frac{1}{\pi n^{2}}2 x \sin{\left(n x \right)}} \right|_{- \pi}^{\pi}-\frac{1}{\pi n^{2}}\int\limits_{- \pi}^{\pi} {2 \sin{\left(n x \right)}dx}=\frac{2}{\pi n^{3}}\left. {\cos{\left(n x \right)}} \right|_{- \pi}^{\pi}=0\]

Таким образом, ряд Фурье:

\[f\left( x \right) = \frac{\pi^{2}}{3}+ \sum\limits_{n = 1}^\infty {{\frac{4 \left(-1\right)^{n}}{n^{2}}}\cos nx}.\]

Сумма ряда Фурье в точках разрыва равна полусумме значений функции слева и справа. В остальных точках - значению функции.

\[S\left( - \pi \right)=\frac{\pi^{2}+\pi^{2}}{2}=\pi^{2},\:S\left( \pi \right)=\frac{\pi^{2}+\pi^{2}}{2}=\pi^{2}\]

$f\left({x}\right)=x^{2},\quad $ \newline Функция удовлетворяет условиях Дирихле и может быть разложена в ряд Фурье. \newline \textbf{Ряд Фурье общего вида для $2\pi$ периодичных функций:} \newline $f\left( x \right) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {\left( {{a_{n}}\cos nx + {b_{n}}\sin nx} \right)}$ \newline Вычислим коэффициенты ряда: \newline $a_0=\frac{1}{\pi}\int\limits_{0}^{\pi}{f\left({x}\right)dx}=\int\limits_{- \pi}^{\pi} {\frac{1}{\pi}\cdot x^{2}dx} =\left.{\frac{x^{3}}{3 \pi}}\right | _{- \pi}^{\pi} =\frac{\pi^{3}}{3 \pi}- \frac{\left(- \pi\right)^{3}}{3 \pi}=\frac{2 \pi^{2}}{3} \approx 6.5797$ \newline $a_{n} = \frac{1}{L}\int\limits_{-\pi}^{\pi} {f\left( x \right)\cos nxdx} =\frac{1}{\pi}\int\limits_{- \pi}^{\pi} {x^{2}\cos{\left(n x \right)}dx}=\left| {\begin{array}{*{20}{c}}u=x^{2}&du=2 xdx\\v=-\frac{1}{\pi n}\sin{\left(n x \right)}&dv=\frac{1}{\pi}\cos{\left(n x \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}x^{2} \sin{\left(n x \right)}} \right|_{- \pi}^{\pi}+\frac{1}{\pi n}\int\limits_{- \pi}^{\pi} {2 x \sin{\left(n x \right)}dx}=- \frac{2 \pi \sin{\left(\pi n \right)}}{n}+\frac{1}{\pi n}\int\limits_{- \pi}^{\pi} {2 x \sin{\left(n x \right)}dx}=\left| {\begin{array}{*{20}{c}}u=2 x&du=2dx\\v=\frac{1}{\pi n^{2}}\cos{\left(n x \right)}&dv=\frac{1}{\pi n}\sin{\left(n x \right)}dx\end{array}} \right| = - \frac{2 \pi \sin{\left(\pi n \right)}}{n}\left. {+\frac{1}{\pi n^{2}}2 x \cos{\left(n x \right)}} \right|_{- \pi}^{\pi}-\frac{1}{\pi n^{2}}\int\limits_{- \pi}^{\pi} {2 \cos{\left(n x \right)}dx}=\frac{- 2 \pi^{2} \sin{\left(\pi n \right)} + 4 \pi \cos{\left(\pi n \right)}}{\pi n^{3}}+\frac{2}{\pi n^{3}}\left. {\sin{\left(n x \right)}} \right|_{- \pi}^{\pi}=\frac{- 2 \pi^{2} \sin{\left(\pi n \right)} + 4 \pi \cos{\left(\pi n \right)}}{\pi n^{2}} + \frac{4 \sin{\left(\pi n \right)}}{\pi n^{3}}=\frac{4 \left(-1\right)^{n}}{n^{2}}$ \newline $b_{n} = \frac{1}{L}\int\limits_{-\pi}^{\pi} {f\left( x \right)\sin nxdx} =\frac{1}{\pi}\int\limits_{- \pi}^{\pi} {x^{2}\sin{\left(n x \right)}dx}=\left| {\begin{array}{*{20}{c}}u=x^{2}&du=2 xdx\\v=-\frac{1}{\pi n}\cos{\left(n x \right)}&dv=\frac{1}{\pi}\sin{\left(n x \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}x^{2} \cos{\left(n x \right)}} \right|_{- \pi}^{\pi}+\frac{1}{\pi n}\int\limits_{- \pi}^{\pi} {2 x \cos{\left(n x \right)}dx}=+\frac{1}{\pi n}\int\limits_{- \pi}^{\pi} {2 x \cos{\left(n x \right)}dx}=\left| {\begin{array}{*{20}{c}}u=2 x&du=2dx\\v=\frac{1}{\pi n^{2}}\sin{\left(n x \right)}&dv=\frac{1}{\pi n}\cos{\left(n x \right)}dx\end{array}} \right| = \left. {\frac{1}{\pi n^{2}}2 x \sin{\left(n x \right)}} \right|_{- \pi}^{\pi}-\frac{1}{\pi n^{2}}\int\limits_{- \pi}^{\pi} {2 \sin{\left(n x \right)}dx}=\frac{2}{\pi n^{3}}\left. {\cos{\left(n x \right)}} \right|_{- \pi}^{\pi}=0$ \newline Таким образом, ряд Фурье: \newline $f\left( x \right) = \frac{\pi^{2}}{3}+ \sum\limits_{n = 1}^\infty {{\frac{4 \left(-1\right)^{n}}{n^{2}}}\cos nx}.$ \newline Сумма ряда Фурье в точках разрыва равна полусумме значений функции слева и справа. В остальных точках - значению функции. \newline $S\left( - \pi \right)=\frac{\pi^{2}+\pi^{2}}{2}=\pi^{2},\:S\left( \pi \right)=\frac{\pi^{2}+\pi^{2}}{2}=\pi^{2}$ \newline \includegraphics{/home/math8/Math8_v/ExternalstorageMath8/cache/img/09415721893310547inputtolatex.png}\newline