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Задать вопрос экспертам

Консультация № 178526

20.05.2010, 19:05

42.84 руб.

0 2 2

Образование

Снова здравствуйте. Ещё такая задача:

Требуется разложить в ряд Фурье на [2,4]

Обсуждение

давно

Гордиенко Андрей Владимирович

Мастер-Эксперт
17387
18353

21.05.2010, 16:38

общий

это ответ

Здравствуйте, Lola.

Вы можете посмотреть решение задачи здесь.

Проверьте выкладки, ведь от ошибок никто не застрахован.

С уважением.

Об авторе:
Facta loquuntur.

Неизвестный

21.05.2010, 17:06

общий

это ответ

Здравствуйте, Lola.

f(x)=a₀/2+[$8721$]_n=1^[$8734$] (a_n* cos(Pi*n*x/L) + b_n*sin(Pi*n*x/L))

L=1 для отрезка [2;4]

a₀=(1/L)*[$8747$]₂⁴ f(x)dx = [$8747$]₂³ (5-x)dx + [$8747$]₃⁴(4-x)dx =(5*x-x²/2)|₂³+(4*x-x²/2)|₃⁴=15-9/2-10+2+16-8-12+9/2=3

[$8747$]x*cos(Pi*n*x)dx=(1/Pi*n)*(x*sin(Pi*n*x)-[$8747$]sin(Pi*n*x)dx)=(1/Pi*n)*(x*sin(Pi*n*x)+(1/Pi*n)*cos(Pi*n*x))
[$8747$]x*sin(Pi*n*x)dx=(1/Pi*n)*(-x*cos(Pi*n*x)+[$8747$]cos(Pi*n*x)dx)=(1/Pi*n)*(-x*cos(Pi*n*x)+(1/Pi*n)*sin(Pi*n*x))
sin(Pi*n)=0,
cos(Pi*n)=(-1)ⁿ, n=1,2,3,...

a_n=(1/L)*[$8747$]₂⁴ f(x)*cos(Pi*n*x)dx = [$8747$]₂³(5-x)*cos(Pi*n*x)dx + [$8747$]₃⁴(4-x)*cos(Pi*n*x)dx=
=(-6*Pi*n*sin(Pi*n)*cos(Pi*n)+2*cos²(Pi*n)-1+8*Pi*n*sin(Pi*n)*cos²(Pi*n)-2*Pi*n*sin(Pi*n)-4*cos³(Pi*n)+3*cos(Pi*n))/(Pi²*n²) -
-(4*Pi*n*sin(Pi*n)*cos²(Pi*n)-Pi*n*sin(Pi*n)-4*cos³(Pi*n)+3*cos(Pi*n)+8*cos⁴(Pi*n)-8*cos²(Pi*n)+1)/(Pi²*n²)=
=(1/(Pi²*n²))*(2-1-4*(-1)ⁿ+3*(-1)ⁿ+4*(-1)ⁿ-3*(-1)ⁿ-8+8-1)=0

b_n=(1/L)*[$8747$]₂⁴ f(x)*sin(Pi*n*x)dx = [$8747$]₂³(5-x)*sin(Pi*n*x)dx + [$8747$]₃⁴(4-x)*sin(Pi*n*x)dx=(6*Pi*n*cos²(Pi*n)-3*Pi*n+2*sin(Pi*n)*cos(Pi*n)-8*Pi*n*cos³(Pi*n)+6*Pi*n*cos(Pi*n)-4*sin(Pi*n)*cos²(Pi*n)+sin(Pi*n)+
+4*Pi*n*cos³(Pi*n)-3*Pi*n*cos(Pi*n)+4*sin(Pi*n)*cos²(Pi*n)-sin(Pi*n)-8*sin(Pi*n)*cos³(Pi*n)+4*sin(Pi*n)*cos(Pi*n))/(Pi²*n²)=
=(6*Pi*n-3*Pi*n-8*(-1)ⁿ*Pi*n+6*(-1)ⁿ*Pi*n+4*(-1)ⁿ*Pi*n-3*(-1)ⁿ*Pi*n)/(Pi²*n²)=(3-(-1)ⁿ)/(Pi*n)

Получим

f(x)=3/2+[$8721$]_n=1^[$8734$] (1/(Pi*n))*(3-(-1)ⁿ)*sin(Pi*n*x)

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