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Здравствуйте! Решая задачу по матфизике получил ответ в виде интеграла. Никак не могу его взять помогите пожалуйста. Просьба пакеты типа Maple не предлагать, ответ знаю, не могу понять как он получается из этого интеграла. T0- константа, i- мнимая единица. Зарание большое спасибо.
aspmed2: Скажите ответ, так будет легче его вывести.
Неизвестный
14.01.2010, 23:52
общий
Да конечно. http://s001.radikal.ru/i196/1001/f1/d8875727a00b.jpg
Там Ф - это функция ошибок(erf)
Неизвестный
15.01.2010, 08:37
общий
Интеграл по ds в http://s59.radikal.ru/i166/1001/ed/5e2dedb2147d.jpg с точностью до множителя - дельта функция от p. И весь интеграл - константа.
Неизвестный
15.01.2010, 11:05
общий
Покажите пожалуйста как у вас получился такой результат?
Неизвестный
15.01.2010, 12:25
общий
aspmed2: Прошу прощения, я конечно ошибся. В круглых скобках стоит преобразование Фурье от функции Хевисайда. F[θ(s)] = i/(p+i0) = πδ(p) + iP(1/p) Я как-то невнимательно посмотрел задание. По моему, если поменять порядок интегрирования все упроститься. Сейчас попробую написать
Неизвестный
15.01.2010, 12:57
общий
aspmed2: Что такое x0 в ответе?
Неизвестный
15.01.2010, 13:33
общий
T0/(2π)∫-∞∞exp(-p2t+ipx)( ∫0∞exp(ips)ds)dp = T0/(2π) ∫0∞ds∫-∞∞exp(-p2t+ip(x-s)) = T0/(2π) ∫0∞ds√(π/t)exp((s-x)2/(4t)) ={(s-x)/2√t=y; ds = 2√tdy} = T0/√π)∫-x/2√t∞exp(-y2dy = T0/2 erfc(-x/2√t) = T0/2 (1 – erf(-x/2√t)) Вот такой ответ получился.
Неизвестный
15.01.2010, 21:36
общий
Влaдимир: Ах, да. Ответ дан уже с подставленными значениями. То есть изначально вместо То былы функция f(s), и надо было посчитать значения при f(s)=Tо при |s|<xo и при f(s)=0 при |x|>xo. Я просто сразу подставил То вместо f(s).
Спасибо.Сейчас посмотрю ваше решение.
Неизвестный
15.01.2010, 22:54
общий
Влaдимир: Подскажите пожалуйста поподробнее как произошел этот переход: "
aspmed2: ∫-∞∞exp(-p2t+ipq) = √(π/t)exp(-q22/(4t)) Это табличный интеграл - преобразование Фурье от exp(-p2t). Взять этот интеграл можно заменой переменных z = p√t - iq/(2√t). После такой замены интеграл превращается в 1/√t∫exp(-z2-q22/(4t)) Строго говоря при такой замене контур интегрирования - прямая в комплексной плоскости, параллельная вещественной оси. С помощью теории функций комплексной переменной можно доказать, что интеграл по такому контуру равен интегралу по вещественной оси.
Неизвестный
18.01.2010, 10:49
общий
Спасибо. Разобрался. надо почаще заглядовать в таблицу интегралов
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