Sr Examen
Integral de (e^(i*y*t))/(1+t^2)dt dt
Solución
Ha introducido
[src]
oo / | | I*y*t | E | ------ dt | 2 | 1 + t | / -oo
$$\int\limits_{-\infty}^{\infty} \frac{e^{t i y}}{t^{2} + 1}\, dt$$
Integral(E^((i*y)*t)/(1 + t^2), (t, -oo, oo))
Respuesta (Indefinida)
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/ / | | | I*y*t | I*t*y | E | e | ------ dt = C + | ------ dt | 2 | 2 | 1 + t | 1 + t | | / /
$$\int \frac{e^{t i y}}{t^{2} + 1}\, dt = C + \int \frac{e^{i t y}}{t^{2} + 1}\, dt$$
Respuesta
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/ / pi \ / pi \ /pi*I \ / pi*I \ |- |- -- + I*Shi(y)|*cosh(y) - |- -- - I*Shi(y)|*cosh(y) + I*|---- + Chi(y)|*sinh(y) - I*|- ---- + Chi(y)|*sinh(y) for y > 0 | \ 2 / \ 2 / \ 2 / \ 2 / | | oo | / | | < | I*t*y | | e | | ------ dt otherwise | | 2 | | 1 + t | | | / \ -oo
$$\begin{cases} - \left(- i \operatorname{Shi}{\left(y \right)} - \frac{\pi}{2}\right) \cosh{\left(y \right)} - \left(i \operatorname{Shi}{\left(y \right)} - \frac{\pi}{2}\right) \cosh{\left(y \right)} - i \left(\operatorname{Chi}\left(y\right) - \frac{i \pi}{2}\right) \sinh{\left(y \right)} + i \left(\operatorname{Chi}\left(y\right) + \frac{i \pi}{2}\right) \sinh{\left(y \right)} & \text{for}\: y > 0 \\\int\limits_{-\infty}^{\infty} \frac{e^{i t y}}{t^{2} + 1}\, dt & \text{otherwise} \end{cases}$$
=
=
/ / pi \ / pi \ /pi*I \ / pi*I \ |- |- -- + I*Shi(y)|*cosh(y) - |- -- - I*Shi(y)|*cosh(y) + I*|---- + Chi(y)|*sinh(y) - I*|- ---- + Chi(y)|*sinh(y) for y > 0 | \ 2 / \ 2 / \ 2 / \ 2 / | | oo | / | | < | I*t*y | | e | | ------ dt otherwise | | 2 | | 1 + t | | | / \ -oo
$$\begin{cases} - \left(- i \operatorname{Shi}{\left(y \right)} - \frac{\pi}{2}\right) \cosh{\left(y \right)} - \left(i \operatorname{Shi}{\left(y \right)} - \frac{\pi}{2}\right) \cosh{\left(y \right)} - i \left(\operatorname{Chi}\left(y\right) - \frac{i \pi}{2}\right) \sinh{\left(y \right)} + i \left(\operatorname{Chi}\left(y\right) + \frac{i \pi}{2}\right) \sinh{\left(y \right)} & \text{for}\: y > 0 \\\int\limits_{-\infty}^{\infty} \frac{e^{i t y}}{t^{2} + 1}\, dt & \text{otherwise} \end{cases}$$
Piecewise((-(-pi/2 + i*Shi(y))*cosh(y) - (-pi/2 - i*Shi(y))*cosh(y) + i*(pi*i/2 + Chi(y))*sinh(y) - i*(-pi*i/2 + Chi(y))*sinh(y), y > 0), (Integral(exp(i*t*y)/(1 + t^2), (t, -oo, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.