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Консультация № 175193
13.12.2009, 15:31
35.00 руб.
0
3
1
Уважаемые эксперты.
Помогите, пожалуйста, в решении следующей задачи.
Разложить в ряд Фурье функцию
f(x)=-3+|x| , -1<x<1
Помогите, пожалуйста, в решении следующей задачи.
Разложить в ряд Фурье функцию
f(x)=-3+|x| , -1<x<1
Обсуждение
Неизвестный
15.12.2009, 06:16
общий
это ответ
Здравствуйте, filins.
f(x)=-3+|x| - функция четная, поэтому она разлогается
На промежутке -1<x<1 функция f(x) представима в виде ряда Фурье вида
f(x) = a0/2 + ∑k=1∞( bksin(πkx) + akcos(πkx)),(1)
где ak = 1/2∫-11f(x)cos(πkx)dx,
bk = 1/2∫-11f(x)sin(πkx)dx.
В силу нечетности функции f(x)коэффициенты bk = 0, а для ak справедлива формула
ak = ∫01f(x)cos(πkx)dx.
Для k = 0
a0 = ∫01(|x|-3)dx = x2|01 – 6x|01 = 1/2 - 3 = -5/2.
При k>0 для ak получаем
ak = ∫01(x-3)cos(πkx)dx = [cos(πkx)/ (πk)2| 01 + x sin(πkx)/ πk|01 – 3sin(πkx)/ πk|01
Откуда
ak = 0 при k=2m
ak = -2/(πk)2, при k=2m+1.
Подставляя выражения для ak и bk в формулу (1) получаем окончательный ответ
|x| - 3 = -5/2 - 2∑m=1∞cos(2m+1) πx/(π(2m+1))2.
f(x)=-3+|x| - функция четная, поэтому она разлогается
На промежутке -1<x<1 функция f(x) представима в виде ряда Фурье вида
f(x) = a0/2 + ∑k=1∞( bksin(πkx) + akcos(πkx)),(1)
где ak = 1/2∫-11f(x)cos(πkx)dx,
bk = 1/2∫-11f(x)sin(πkx)dx.
В силу нечетности функции f(x)коэффициенты bk = 0, а для ak справедлива формула
ak = ∫01f(x)cos(πkx)dx.
Для k = 0
a0 = ∫01(|x|-3)dx = x2|01 – 6x|01 = 1/2 - 3 = -5/2.
При k>0 для ak получаем
ak = ∫01(x-3)cos(πkx)dx = [cos(πkx)/ (πk)2| 01 + x sin(πkx)/ πk|01 – 3sin(πkx)/ πk|01
Откуда
ak = 0 при k=2m
ak = -2/(πk)2, при k=2m+1.
Подставляя выражения для ak и bk в формулу (1) получаем окончательный ответ
|x| - 3 = -5/2 - 2∑m=1∞cos(2m+1) πx/(π(2m+1))2.
5
Форма ответа
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