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Консультация № 185871
21.04.2012, 22:48
62.88 руб.
0
2
2
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Вычислить интеграл при помощи теоремы вычетов:
Заранее благодарен за помощь!
Вычислить интеграл при помощи теоремы вычетов:
Заранее благодарен за помощь!
Обсуждение
21.04.2012, 23:33
общий
это ответ
Здравствуйте, G-buck!
Интеграл равен 2*Pi*i, умноженному на сумму вычетов в точках z=0 и z=Pi.
Обе точки являются полюсами второго порядка.
Вычет f(z) в полюсе z=a второго порядка находим по формуле
res f(z)=[(z-a)2f(z)]'|z=a
В точке z=0
resf(z)=[cos z/(z-Pi)2]'|z=0=[-(sin z)(z-Pi)2-(cos z)*2(z-Pi)]/(z-Pi)4|z=0=2/Pi3
В точке z=Pi
resf(z)=[cos z/z2]'|z=Pi=[-(sin z)z2-(cos z)*2z)]/z4|z=Pi=2/Pi3
Интеграл равен 2*Pi*i*(4/Pi3)=8i/Pi2
Интеграл равен 2*Pi*i, умноженному на сумму вычетов в точках z=0 и z=Pi.
Обе точки являются полюсами второго порядка.
Вычет f(z) в полюсе z=a второго порядка находим по формуле
res f(z)=[(z-a)2f(z)]'|z=a
В точке z=0
resf(z)=[cos z/(z-Pi)2]'|z=0=[-(sin z)(z-Pi)2-(cos z)*2(z-Pi)]/(z-Pi)4|z=0=2/Pi3
В точке z=Pi
resf(z)=[cos z/z2]'|z=Pi=[-(sin z)z2-(cos z)*2z)]/z4|z=Pi=2/Pi3
Интеграл равен 2*Pi*i*(4/Pi3)=8i/Pi2
5
21.04.2012, 23:37
общий
это ответ
Здравствуйте, G-buck!
Подынтегральная функция голоморфна в указанном круге, за исключением двух точек: z = 0 и z =
. Поэтому искомый интеграл равен сумме вычетов, умноженных на 
:
.
Вычеты равны (-1)-му коэффициенту разложения в ряд Лорана. Вот не мудрствуя лукаво и найдём коэффициенты в данных точках.
В нуле:
.
Т.е.
.
В пи:
.
Т.е.
.
Итак, значение интеграла есть
.
Подынтегральная функция голоморфна в указанном круге, за исключением двух точек: z = 0 и z =
. Поэтому искомый интеграл равен сумме вычетов, умноженных на :.Вычеты равны (-1)-му коэффициенту разложения в ряд Лорана. Вот не мудрствуя лукаво и найдём коэффициенты в данных точках.
В нуле:
.Т.е.
.В пи:
.Т.е.
.Итак, значение интеграла есть
.
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