Sr Examen
Integral de exp((-j)*w*c)*a^2*exp(-b*|c|) dc
Solución
Ha introducido
[src]
oo / | | -I*w*c 2 -b*|c| | e *a *e dc | / -oo
$$\int\limits_{-\infty}^{\infty} a^{2} e^{c - i w} e^{- b \left|{c}\right|}\, dc$$
Integral((exp(((-i)*w)*c)*a^2)*exp((-b)*|c|), (c, -oo, oo))
Respuesta (Indefinida)
[src]
/ / | | | -I*w*c 2 -b*|c| 2 | -b*|c| -I*w*c | e *a *e dc = C + a * | e *e dc | | / /
$$\int a^{2} e^{c - i w} e^{- b \left|{c}\right|}\, dc = C + a^{2} \int e^{- b \left|{c}\right|} e^{c - i w}\, dc$$
Respuesta
[src]
/ 2 2 | a a / / / pi | pi | pi\ /| pi | pi pi\ /| pi | pi pi\\ / / pi |pi | pi\ /|pi | pi pi\ /|pi | pi pi\\\ |----------- + ----------- for And|Or|And||arg(b)| <= --, |- -- + arg(w)| < --|, And||- -- + arg(w)| <= --, |arg(b)| < --|, And||- -- + arg(w)| < --, |arg(b)| < --||, Or|And||arg(b)| <= --, |-- + arg(w)| < --|, And||-- + arg(w)| <= --, |arg(b)| < --|, And||-- + arg(w)| < --, |arg(b)| < --||| | / I*w\ / I*w\ \ \ \ 2 | 2 | 2 / \| 2 | 2 2 / \| 2 | 2 2 // \ \ 2 |2 | 2 / \|2 | 2 2 / \|2 | 2 2 /// |b*|1 + ---| b*|1 - ---| | \ b / \ b / | < oo | / | | | | 2 -b*|c| -I*c*w | | a *e *e dc otherwise | | |/ \-oo
$$\begin{cases} \frac{a^{2}}{b \left(1 + \frac{i w}{b}\right)} + \frac{a^{2}}{b \left(1 - \frac{i w}{b}\right)} & \text{for}\: \left(\left(\left|{\arg{\left(b \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(w \right)} - \frac{\pi}{2}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(w \right)} - \frac{\pi}{2}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(b \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(w \right)} - \frac{\pi}{2}}\right| < \frac{\pi}{2} \wedge \left|{\arg{\left(b \right)}}\right| < \frac{\pi}{2}\right)\right) \wedge \left(\left(\left|{\arg{\left(b \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(w \right)} + \frac{\pi}{2}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(w \right)} + \frac{\pi}{2}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(b \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(w \right)} + \frac{\pi}{2}}\right| < \frac{\pi}{2} \wedge \left|{\arg{\left(b \right)}}\right| < \frac{\pi}{2}\right)\right) \\\int\limits_{-\infty}^{\infty} a^{2} e^{- b \left|{c}\right|} e^{- i c w}\, dc & \text{otherwise} \end{cases}$$
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/ 2 2 | a a / / / pi | pi | pi\ /| pi | pi pi\ /| pi | pi pi\\ / / pi |pi | pi\ /|pi | pi pi\ /|pi | pi pi\\\ |----------- + ----------- for And|Or|And||arg(b)| <= --, |- -- + arg(w)| < --|, And||- -- + arg(w)| <= --, |arg(b)| < --|, And||- -- + arg(w)| < --, |arg(b)| < --||, Or|And||arg(b)| <= --, |-- + arg(w)| < --|, And||-- + arg(w)| <= --, |arg(b)| < --|, And||-- + arg(w)| < --, |arg(b)| < --||| | / I*w\ / I*w\ \ \ \ 2 | 2 | 2 / \| 2 | 2 2 / \| 2 | 2 2 // \ \ 2 |2 | 2 / \|2 | 2 2 / \|2 | 2 2 /// |b*|1 + ---| b*|1 - ---| | \ b / \ b / | < oo | / | | | | 2 -b*|c| -I*c*w | | a *e *e dc otherwise | | |/ \-oo
$$\begin{cases} \frac{a^{2}}{b \left(1 + \frac{i w}{b}\right)} + \frac{a^{2}}{b \left(1 - \frac{i w}{b}\right)} & \text{for}\: \left(\left(\left|{\arg{\left(b \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(w \right)} - \frac{\pi}{2}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(w \right)} - \frac{\pi}{2}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(b \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(w \right)} - \frac{\pi}{2}}\right| < \frac{\pi}{2} \wedge \left|{\arg{\left(b \right)}}\right| < \frac{\pi}{2}\right)\right) \wedge \left(\left(\left|{\arg{\left(b \right)}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(w \right)} + \frac{\pi}{2}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(w \right)} + \frac{\pi}{2}}\right| \leq \frac{\pi}{2} \wedge \left|{\arg{\left(b \right)}}\right| < \frac{\pi}{2}\right) \vee \left(\left|{\arg{\left(w \right)} + \frac{\pi}{2}}\right| < \frac{\pi}{2} \wedge \left|{\arg{\left(b \right)}}\right| < \frac{\pi}{2}\right)\right) \\\int\limits_{-\infty}^{\infty} a^{2} e^{- b \left|{c}\right|} e^{- i c w}\, dc & \text{otherwise} \end{cases}$$
Piecewise((a^2/(b*(1 + i*w/b)) + a^2/(b*(1 - i*w/b)), (((Abs(arg(b)) <= pi/2)∧(Abs(-pi/2 + arg(w)) < pi/2))∨((Abs(arg(b)) < pi/2)∧(Abs(-pi/2 + arg(w)) <= pi/2))∨((Abs(arg(b)) < pi/2)∧(Abs(-pi/2 + arg(w)) < pi/2)))∧(((Abs(arg(b)) <= pi/2)∧(Abs(pi/2 + arg(w)) < pi/2))∨((Abs(arg(b)) < pi/2)∧(Abs(pi/2 + arg(w)) <= pi/2))∨((Abs(arg(b)) < pi/2)∧(Abs(pi/2 + arg(w)) < pi/2)))), (Integral(a^2*exp(-b*|c|)*exp(-i*c*w), (c, -oo, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.