\[f\left({x}\right)=\begin{cases}x,& -2 < x <0,\\\pi - x,& 0 < x <2\end{cases},\quad L=2\]

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

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

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

Вычислим коэффициенты ряда:

\[a_0=\frac{1}{L}\int\limits_{0}^{L}{f\left({x}\right)dx}=\int\limits_{-2}^{0} {\frac{1}{2}\cdot xdx} +\int\limits_{0}^{2} {\frac{1}{2}\cdot \left( \pi - x \right)dx} =\left.{\frac{x^{2}}{4}}\right | _{-2}^{0} +\left.{\left( - \frac{x^{2}}{4} + \frac{\pi x}{2} \right)}\right | _{0}^{2} =\frac{0^{2}}{4}- \frac{\left(-2\right)^{2}}{4}- \frac{2^{2}}{4} + \frac{\pi \cdot 2}{2}+\frac{0^{2}}{4} - \frac{\pi \cdot 0}{2}=-2 + \pi \approx 1.1416\]

\[•\:a_{n}= \frac{1}{L}\int\limits_{-L}^{L} {f\left( x \right)\cos \frac{{n\pi x}}{L}dx} =\frac{1}{2}\int\limits_{-2}^{0}{x\cos {\frac{\pi n x}{2}}dx}+\frac{1}{2}\int\limits_{0}^{2}{\left( \pi - x \right)\cos {\frac{\pi n x}{2}}dx}\]

Вычислим отдельно:

\[\frac{1}{2}\int\limits_{-2}^{0} {x\cos{\left(\frac{\pi n x}{2} \right)}dx}=\left| {\begin{array}{*{20}{c}}u=x&du=dx\\v=-\frac{1}{\pi n}\sin{\left(\frac{\pi n x}{2} \right)}&dv=\frac{1}{2}\cos{\left(\frac{\pi n x}{2} \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}x \sin{\left(\frac{\pi n x}{2} \right)}} \right|_{-2}^{0}+\frac{1}{\pi n}\int\limits_{-2}^{0} {\sin{\left(\frac{\pi n x}{2} \right)}dx}=\frac{4 \sin{\left(\pi n \right)}}{\pi^{2} n^{2}}+\frac{2}{\pi^{2} n^{2}}\left. {\cos{\left(\frac{\pi n x}{2} \right)}} \right|_{-2}^{0}=\frac{2 \sin{\left(\pi n \right)}}{\pi n} + \frac{2 \cdot \left(1 - \cos{\left(\pi n \right)}\right)}{\pi^{2} n^{2}}=- \frac{2 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{2}{\pi^{2} n^{2}}\]

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

Тогда

\[a_{n}=- \frac{2 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{2}{\pi^{2} n^{2}}- \frac{2 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{2}{\pi^{2} n^{2}}=- \frac{4 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{4}{\pi^{2} n^{2}}\]

\[•\:b_{n}= \frac{1}{L}\int\limits_{-L}^{L} {f\left( x \right)\sin \frac{{n\pi x}}{L}dx} =\frac{1}{2}\int\limits_{-2}^{0}{x\sin {\frac{\pi n x}{2}}dx}+\frac{1}{2}\int\limits_{0}^{2}{\left( \pi - x \right)\sin {\frac{\pi n x}{2}}dx}\]

Вычислим отдельно:

\[\frac{1}{2}\int\limits_{-2}^{0} {x\sin{\left(\frac{\pi n x}{2} \right)}dx}=\left| {\begin{array}{*{20}{c}}u=x&du=dx\\v=-\frac{1}{\pi n}\cos{\left(\frac{\pi n x}{2} \right)}&dv=\frac{1}{2}\sin{\left(\frac{\pi n x}{2} \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}x \cos{\left(\frac{\pi n x}{2} \right)}} \right|_{-2}^{0}+\frac{1}{\pi n}\int\limits_{-2}^{0} {\cos{\left(\frac{\pi n x}{2} \right)}dx}=- \frac{4 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}}+\frac{2}{\pi^{2} n^{2}}\left. {\sin{\left(\frac{\pi n x}{2} \right)}} \right|_{-2}^{0}=- \frac{2 \cos{\left(\pi n \right)}}{\pi n} + \frac{2 \sin{\left(\pi n \right)}}{\pi^{2} n^{2}}=- \frac{2 \left(-1\right)^{n}}{\pi n}\]

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

Тогда

\[b_{n}=- \frac{2 \left(-1\right)^{n}}{\pi n}- \frac{\left(-1\right)^{n}}{n} + \frac{2 \left(-1\right)^{n}}{\pi n} + \frac{1}{n}=- \frac{\left(-1\right)^{n}}{n} + \frac{1}{n}\]

Таким образом, ряд Фурье:

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

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

\[S\left( -2 \right)=\frac{-2 + \pi+\left( -2 \right)}{2}=-2 + \frac{\pi}{2},\:S\left( 0 \right)=\frac{0+\pi}{2}=\frac{\pi}{2},\:S\left( 2 \right)=\frac{-2 + \pi+\left( -2 \right)}{2}=-2 + \frac{\pi}{2}\]

$f\left({x}\right)=\begin{cases}x,& -2 < x <0,\\\pi - x,& 0 < x <2\end{cases},\quad L=2$ \newline Функция удовлетворяет условиях Дирихле и может быть разложена в ряд Фурье. \newline \textbf{Ряд Фурье общего вида для $2L$ периодичных функций:} \newline $f\left( x \right) = \frac{{{a_0}}}{2} + \sum\limits_{n = 1}^\infty {\left( {{a_{n}}\cos \frac{n\pi x}{L} + {b_{n}}\sin \frac{n\pi x}{L}} \right)}$ \newline Вычислим коэффициенты ряда: \newline $a_0=\frac{1}{L}\int\limits_{0}^{L}{f\left({x}\right)dx}=\int\limits_{-2}^{0} {\frac{1}{2}\cdot xdx} +\int\limits_{0}^{2} {\frac{1}{2}\cdot \left( \pi - x \right)dx} =\left.{\frac{x^{2}}{4}}\right | _{-2}^{0} +\left.{\left( - \frac{x^{2}}{4} + \frac{\pi x}{2} \right)}\right | _{0}^{2} =\frac{0^{2}}{4}- \frac{\left(-2\right)^{2}}{4}- \frac{2^{2}}{4} + \frac{\pi \cdot 2}{2}+\frac{0^{2}}{4} - \frac{\pi \cdot 0}{2}=-2 + \pi \approx 1.1416$ \newline $•\:a_{n}= \frac{1}{L}\int\limits_{-L}^{L} {f\left( x \right)\cos \frac{{n\pi x}}{L}dx} =\frac{1}{2}\int\limits_{-2}^{0}{x\cos {\frac{\pi n x}{2}}dx}+\frac{1}{2}\int\limits_{0}^{2}{\left( \pi - x \right)\cos {\frac{\pi n x}{2}}dx}$ \newline Вычислим отдельно: \newline $\frac{1}{2}\int\limits_{-2}^{0} {x\cos{\left(\frac{\pi n x}{2} \right)}dx}=\left| {\begin{array}{*{20}{c}}u=x&du=dx\\v=-\frac{1}{\pi n}\sin{\left(\frac{\pi n x}{2} \right)}&dv=\frac{1}{2}\cos{\left(\frac{\pi n x}{2} \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}x \sin{\left(\frac{\pi n x}{2} \right)}} \right|_{-2}^{0}+\frac{1}{\pi n}\int\limits_{-2}^{0} {\sin{\left(\frac{\pi n x}{2} \right)}dx}=\frac{4 \sin{\left(\pi n \right)}}{\pi^{2} n^{2}}+\frac{2}{\pi^{2} n^{2}}\left. {\cos{\left(\frac{\pi n x}{2} \right)}} \right|_{-2}^{0}=\frac{2 \sin{\left(\pi n \right)}}{\pi n} + \frac{2 \cdot \left(1 - \cos{\left(\pi n \right)}\right)}{\pi^{2} n^{2}}=- \frac{2 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{2}{\pi^{2} n^{2}}$ \newline $\frac{1}{2}\int\limits_{0}^{2} {\left( \pi - x \right)\cos{\left(\frac{\pi n x}{2} \right)}dx}=\left| {\begin{array}{*{20}{c}}u=\pi - x&du=-dx\\v=-\frac{1}{\pi n}\sin{\left(\frac{\pi n x}{2} \right)}&dv=\frac{1}{2}\cos{\left(\frac{\pi n x}{2} \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}\left(\pi - x\right) \sin{\left(\frac{\pi n x}{2} \right)}} \right|_{0}^{2}-\frac{1}{\pi n}\int\limits_{0}^{2} {\sin{\left(\frac{\pi n x}{2} \right)}dx}=- \frac{2 \left(-2 + \pi\right) \sin{\left(\pi n \right)}}{\pi^{2} n^{2}}-\frac{2}{\pi^{2} n^{2}}\left. {\cos{\left(\frac{\pi n x}{2} \right)}} \right|_{0}^{2}=- \frac{\left(-2 + \pi\right) \sin{\left(\pi n \right)}}{\pi n} + \frac{2 \cdot \left(1 - \cos{\left(\pi n \right)}\right)}{\pi^{2} n^{2}}=- \frac{2 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{2}{\pi^{2} n^{2}}$ \newline Тогда \newline $a_{n}=- \frac{2 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{2}{\pi^{2} n^{2}}- \frac{2 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{2}{\pi^{2} n^{2}}=- \frac{4 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{4}{\pi^{2} n^{2}}$ \newline $•\:b_{n}= \frac{1}{L}\int\limits_{-L}^{L} {f\left( x \right)\sin \frac{{n\pi x}}{L}dx} =\frac{1}{2}\int\limits_{-2}^{0}{x\sin {\frac{\pi n x}{2}}dx}+\frac{1}{2}\int\limits_{0}^{2}{\left( \pi - x \right)\sin {\frac{\pi n x}{2}}dx}$ \newline Вычислим отдельно: \newline $\frac{1}{2}\int\limits_{-2}^{0} {x\sin{\left(\frac{\pi n x}{2} \right)}dx}=\left| {\begin{array}{*{20}{c}}u=x&du=dx\\v=-\frac{1}{\pi n}\cos{\left(\frac{\pi n x}{2} \right)}&dv=\frac{1}{2}\sin{\left(\frac{\pi n x}{2} \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}x \cos{\left(\frac{\pi n x}{2} \right)}} \right|_{-2}^{0}+\frac{1}{\pi n}\int\limits_{-2}^{0} {\cos{\left(\frac{\pi n x}{2} \right)}dx}=- \frac{4 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}}+\frac{2}{\pi^{2} n^{2}}\left. {\sin{\left(\frac{\pi n x}{2} \right)}} \right|_{-2}^{0}=- \frac{2 \cos{\left(\pi n \right)}}{\pi n} + \frac{2 \sin{\left(\pi n \right)}}{\pi^{2} n^{2}}=- \frac{2 \left(-1\right)^{n}}{\pi n}$ \newline $\frac{1}{2}\int\limits_{0}^{2} {\left( \pi - x \right)\sin{\left(\frac{\pi n x}{2} \right)}dx}=\left| {\begin{array}{*{20}{c}}u=\pi - x&du=-dx\\v=-\frac{1}{\pi n}\cos{\left(\frac{\pi n x}{2} \right)}&dv=\frac{1}{2}\sin{\left(\frac{\pi n x}{2} \right)}dx\end{array}} \right| = \left. {-\frac{1}{\pi n}\left(\pi - x\right) \cos{\left(\frac{\pi n x}{2} \right)}} \right|_{0}^{2}-\frac{1}{\pi n}\int\limits_{0}^{2} {\cos{\left(\frac{\pi n x}{2} \right)}dx}=\frac{2 \left(- \left(-2 + \pi\right) \cos{\left(\pi n \right)} + \pi\right)}{\pi^{2} n^{2}}-\frac{2}{\pi^{2} n^{2}}\left. {\sin{\left(\frac{\pi n x}{2} \right)}} \right|_{0}^{2}=\frac{- \left(-2 + \pi\right) \cos{\left(\pi n \right)} + \pi}{\pi n} - \frac{2 \sin{\left(\pi n \right)}}{\pi^{2} n^{2}}=- \frac{\left(-1\right)^{n}}{n} + \frac{2 \left(-1\right)^{n}}{\pi n} + \frac{1}{n}$ \newline Тогда \newline $b_{n}=- \frac{2 \left(-1\right)^{n}}{\pi n}- \frac{\left(-1\right)^{n}}{n} + \frac{2 \left(-1\right)^{n}}{\pi n} + \frac{1}{n}=- \frac{\left(-1\right)^{n}}{n} + \frac{1}{n}$ \newline Таким образом, ряд Фурье: \newline $f\left( x \right) = -1 + \frac{\pi}{2}+ \sum\limits_{n = 1}^\infty {\left( {{\left( - \frac{4 \left(-1\right)^{n}}{\pi^{2} n^{2}} + \frac{4}{\pi^{2} n^{2}} \right)}\cos \frac{\pi n x}{2}- {\left( \frac{\left(-1\right)^{n}}{n} - \frac{1}{n} \right)}\sin \frac{\pi n x}{2}} \right)}.$ \newline Сумма ряда Фурье в точках разрыва равна полусумме значений функции слева и справа. В остальных точках - значению функции. \newline $S\left( -2 \right)=\frac{-2 + \pi+\left( -2 \right)}{2}=-2 + \frac{\pi}{2},\:S\left( 0 \right)=\frac{0+\pi}{2}=\frac{\pi}{2},\:S\left( 2 \right)=\frac{-2 + \pi+\left( -2 \right)}{2}=-2 + \frac{\pi}{2}$ \newline \includegraphics{/home/math8/Math8_v/ExternalstorageMath8/cache/img/9123389720916748inputtolatex.png}\newline