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Консультация № 181392
16.12.2010, 22:07
51.37 руб.
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Уважаемые знатоки! Не могли бы вы помочь в решении задачи по функциональному анализу:
Обсуждение
17.12.2010, 15:43
общий
это ответ
Здравствуйте, Мария!
В данном случае t^n - бесконечно дифференцируемая функция, δ(t) - обобщенная функция, поэтому
(t·δ)' = t'·δ + t·δ' = δ - δ = 0, так как t·δ'(t) = -δ(t);
(t^2·δ)' = 2t·δ + t^2·δ' = 2t·δ - t·δ = t·δ
(t^2·δ)" = (t·δ)' = 0.
В общем случае
(t^n·δ)' = nt^(n-1)·δ + t^n·δ' = nt^(n-1)·δ - t^(n-1)·δ = (n-1)t^(n-1)·δ;
(t^n·δ)" = ((n-1)t^(n-1)·δ)' = (n-1)^2t^(n-2)·δ + (n-1)t^(n-1)·δ' = (n-1)^2t^(n-2)·δ - (n-1)t^(n-2)·δ = (n-1)(n-2)t^(n-2)·δ;
(t^n·δ)"' = ((n-1)(n-2)t^(n-2)·δ)' = (n-1)(n-2)^2t^(n-3)·δ + (n-1)(n-2)t^(n-2)·δ' = (n-1)(n-2)^2t^(n-3)·δ - (n-1)(n-2)t^(n-3)·δ = (n-1)(n-2)(n-3)t^(n-3)·δ;
Производная (n-1)-го порядка будет равна (n-1)!t·δ, а производная n-го порядка - 0.
В данном случае t^n - бесконечно дифференцируемая функция, δ(t) - обобщенная функция, поэтому
(t·δ)' = t'·δ + t·δ' = δ - δ = 0, так как t·δ'(t) = -δ(t);
(t^2·δ)' = 2t·δ + t^2·δ' = 2t·δ - t·δ = t·δ
(t^2·δ)" = (t·δ)' = 0.
В общем случае
(t^n·δ)' = nt^(n-1)·δ + t^n·δ' = nt^(n-1)·δ - t^(n-1)·δ = (n-1)t^(n-1)·δ;
(t^n·δ)" = ((n-1)t^(n-1)·δ)' = (n-1)^2t^(n-2)·δ + (n-1)t^(n-1)·δ' = (n-1)^2t^(n-2)·δ - (n-1)t^(n-2)·δ = (n-1)(n-2)t^(n-2)·δ;
(t^n·δ)"' = ((n-1)(n-2)t^(n-2)·δ)' = (n-1)(n-2)^2t^(n-3)·δ + (n-1)(n-2)t^(n-2)·δ' = (n-1)(n-2)^2t^(n-3)·δ - (n-1)(n-2)t^(n-3)·δ = (n-1)(n-2)(n-3)t^(n-3)·δ;
Производная (n-1)-го порядка будет равна (n-1)!t·δ, а производная n-го порядка - 0.
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