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Resolución de integrales/ (e^(i*y*t))/(1+t^2)dt

Integral de (e^(i*y*t))/(1+t^2)dt dt

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 oo          
  /          
 |           
 |   I*y*t   
 |  E        
 |  ------ dt
 |       2   
 |  1 + t    
 |           
/            
-oo
∫−∞∞etiyt2+1 dt\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) [src] 
  /                  /         
 |                  |          
 |  I*y*t           |  I*t*y   
 | E                | e        
 | ------ dt = C +  | ------ dt
 |      2           |      2   
 | 1 + t            | 1 + t    
 |                  |          
/                  /
∫etiyt2+1 dt=C+∫eityt2+1 dt\int \frac{e^{t i y}}{t^{2} + 1}\, dt = C + \int \frac{e^{i t y}}{t^{2} + 1}\, dt
Simplificar
Respuesta [src] 
/  /  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
{−(−iShi⁡(y)−π2)cosh⁡(y)−(iShi⁡(y)−π2)cosh⁡(y)−i(Chi⁡(y)−iπ2)sinh⁡(y)+i(Chi⁡(y)+iπ2)sinh⁡(y)for y>0∫−∞∞eityt2+1 dtotherwise\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
{−(−iShi⁡(y)−π2)cosh⁡(y)−(iShi⁡(y)−π2)cosh⁡(y)−i(Chi⁡(y)−iπ2)sinh⁡(y)+i(Chi⁡(y)+iπ2)sinh⁡(y)for y>0∫−∞∞eityt2+1 dtotherwise\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.

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