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Консультация № 183265
21.05.2011, 00:11
46.42 руб.
0
1
1
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Представить на интервале от (0,[$8719$]) в виде тригонометрического ряда Фурье по косинусу след. функцию
g(x) = x cos(11x).
Заранее огромное спасибо!
Представить на интервале от (0,[$8719$]) в виде тригонометрического ряда Фурье по косинусу след. функцию
g(x) = x cos(11x).
Заранее огромное спасибо!
Обсуждение
21.05.2011, 09:35
общий
это ответ
Здравствуйте, John_the_Revelator!
Коэффициенты Фурье
an=(2/Pi)[$8747$]0Pix*cos11x*cos nx dx=
=(1/Pi)[$8747$]0Pix(cos(n-11)x+cos(n+11)x)dx
Если n[$8800$]11, то
an=(1/Pi)[(xsin(n-11)x/(n-11)+cos(n-11)x/(n-11)2)+((xsin(n+11)x/(n+11)+cos(n+11)x/(n+11)2)]0Pi=
=(1/Pi)[((-1)n+1-1)/(n-11)2+((-1)n+1-1)/(n+11)2]=2((-1)n+1-1)(n2+121)/Pi(n2-121)2
Если n=11, то
a11=(1/Pi)[$8747$]0Pix(1+cos22x)dx=(1/Pi)(x2/2+(xsin22x/22+cos22x/484)|0Pi=Pi/2
Таким образом,
f(x)=Pi/4+(2/Pi)[$8721$]n=1[$8734$][((-1)n+1-1)(n2+121)/(n2-121)2]cos nx
Коэффициенты Фурье
an=(2/Pi)[$8747$]0Pix*cos11x*cos nx dx=
=(1/Pi)[$8747$]0Pix(cos(n-11)x+cos(n+11)x)dx
Если n[$8800$]11, то
an=(1/Pi)[(xsin(n-11)x/(n-11)+cos(n-11)x/(n-11)2)+((xsin(n+11)x/(n+11)+cos(n+11)x/(n+11)2)]0Pi=
=(1/Pi)[((-1)n+1-1)/(n-11)2+((-1)n+1-1)/(n+11)2]=2((-1)n+1-1)(n2+121)/Pi(n2-121)2
Если n=11, то
a11=(1/Pi)[$8747$]0Pix(1+cos22x)dx=(1/Pi)(x2/2+(xsin22x/22+cos22x/484)|0Pi=Pi/2
Таким образом,
f(x)=Pi/4+(2/Pi)[$8721$]n=1[$8734$][((-1)n+1-1)(n2+121)/(n2-121)2]cos nx
Форма ответа
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