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Консультация № 187822
21.04.2014, 17:37
99.36 руб.
0
3
1
Здравствуйте, уважаемые эксперты! Прошу вас помочь разложить функцию y = |x| в ряд Фурье на [-2; +2] и во вторую очередь, если у кого-нибудь будет желание, прошу помочь с решением системы дифференциальных уравнений, которую вы можете увидеть на прикрепленной фотографии, там есть первая часть его решения: http://f-lite.ru/lfp/s020.radikal.ru/i706/1404/90/56dd25bded93.jpg/htm
Обсуждение
21.04.2014, 22:49
общий
это ответ
Здравствуйте, Андрей!
Функция y=|x| является чётной. Коэффициенты Фурье чётной функции f(x) на отрезке [-l ; l] находятся по формулам:
a0=(2/l)[$183$]0[i]l[/i][$8747$]f(x)dx , an=(2/l)[$183$]0[i]l[/i][$8747$]f(x)cos(n[$960$]x/l)dx, bn=0 (n=1,2,...).
Подставляя l=2 и f(x)=|x|, находим
a0=(2/2)[$183$]02[$8747$]xdx= x2/2 |02= 22/2=2,
an=(2/2)[$183$]02[$8747$]xcos(n[$960$]x/2)dx= (2/(n[$960$]))[$183$]02[$8747$]xd(sin(n[$960$]x/2))=(2/(n[$960$]))[$183$]xsin(n[$960$]x/2) |02 - (2/(n[$960$]))[$183$]02[$8747$]sin(n[$960$]x/2)dx=2/(n[$960$])[$183$](sin(n[$960$])-sin0)+(2/(n[$960$]))2[$183$]cos(n[$960$]x/2))|02=
=2/(n[$960$])[$183$](0-0)+(2/(n[$960$]))2[$183$](cos(n[$960$])-cos0)=(2/(n[$960$]))2[$183$]((-1)n-1)=4((-1)n-1)/(n2[$960$]2).
Искомое разложение в ряд Фурье функции y=|x| на [-2 ; 2] имеет вид:
|x|=a0/2+ n=1[$8734$][$8721$](an[$183$]cos(n[$960$]x/2))=2/2+ n=1[$8734$][$8721$](4((-1)n-1)/(n2[$960$]2)[$183$]cos(n[$960$]x/2))=1-(8/pi2)[$183$]cos([$960$]x/2)-(8/(9pi2))[$183$]cos(3[$960$]x/2)-(8/(25pi2))[$183$]cos(5[$960$]x/2)-...
Функция y=|x| является чётной. Коэффициенты Фурье чётной функции f(x) на отрезке [-l ; l] находятся по формулам:
a0=(2/l)[$183$]0[i]l[/i][$8747$]f(x)dx , an=(2/l)[$183$]0[i]l[/i][$8747$]f(x)cos(n[$960$]x/l)dx, bn=0 (n=1,2,...).
Подставляя l=2 и f(x)=|x|, находим
a0=(2/2)[$183$]02[$8747$]xdx= x2/2 |02= 22/2=2,
an=(2/2)[$183$]02[$8747$]xcos(n[$960$]x/2)dx= (2/(n[$960$]))[$183$]02[$8747$]xd(sin(n[$960$]x/2))=(2/(n[$960$]))[$183$]xsin(n[$960$]x/2) |02 - (2/(n[$960$]))[$183$]02[$8747$]sin(n[$960$]x/2)dx=2/(n[$960$])[$183$](sin(n[$960$])-sin0)+(2/(n[$960$]))2[$183$]cos(n[$960$]x/2))|02=
=2/(n[$960$])[$183$](0-0)+(2/(n[$960$]))2[$183$](cos(n[$960$])-cos0)=(2/(n[$960$]))2[$183$]((-1)n-1)=4((-1)n-1)/(n2[$960$]2).
Искомое разложение в ряд Фурье функции y=|x| на [-2 ; 2] имеет вид:
|x|=a0/2+ n=1[$8734$][$8721$](an[$183$]cos(n[$960$]x/2))=2/2+ n=1[$8734$][$8721$](4((-1)n-1)/(n2[$960$]2)[$183$]cos(n[$960$]x/2))=1-(8/pi2)[$183$]cos([$960$]x/2)-(8/(9pi2))[$183$]cos(3[$960$]x/2)-(8/(25pi2))[$183$]cos(5[$960$]x/2)-...
5
Спасибо
Неизвестный
23.04.2014, 22:08
общий
Посмотрите, а на графике я правильно изобразил? http://f-lite.ru/lfp/s018.radikal.ru/i515/1404/59/68247e0f67c4.jpg/htm
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