Integral de exp((-j)*w*c)*a^2*exp(-b*|c|) dc
Solución
Respuesta (Indefinida)
[src]
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| -I*w*c 2 -b*|c| 2 | -b*|c| -I*w*c
| e *a *e dc = C + a * | e *e dc
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∫ a 2 e c − i w e − b ∣ c ∣ d c = C + a 2 ∫ e − b ∣ c ∣ e c − i w d c \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 ∫ a 2 e c − i w e − b ∣ c ∣ d c = C + a 2 ∫ e − b ∣ c ∣ e c − i w d c
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| 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 /
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< oo
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| | 2 -b*|c| -I*c*w
| | a *e *e dc otherwise
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\-oo
{ a 2 b ( 1 + i w b ) + a 2 b ( 1 − i w b ) for ( ( ∣ arg ( b ) ∣ ≤ π 2 ∧ ∣ arg ( w ) − π 2 ∣ < π 2 ) ∨ ( ∣ arg ( w ) − π 2 ∣ ≤ π 2 ∧ ∣ arg ( b ) ∣ < π 2 ) ∨ ( ∣ arg ( w ) − π 2 ∣ < π 2 ∧ ∣ arg ( b ) ∣ < π 2 ) ) ∧ ( ( ∣ arg ( b ) ∣ ≤ π 2 ∧ ∣ arg ( w ) + π 2 ∣ < π 2 ) ∨ ( ∣ arg ( w ) + π 2 ∣ ≤ π 2 ∧ ∣ arg ( b ) ∣ < π 2 ) ∨ ( ∣ arg ( w ) + π 2 ∣ < π 2 ∧ ∣ arg ( b ) ∣ < π 2 ) ) ∫ − ∞ ∞ a 2 e − b ∣ c ∣ e − i c w d c otherwise \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} ⎩ ⎨ ⎧ b ( 1 + b i w ) a 2 + b ( 1 − b i w ) a 2 − ∞ ∫ ∞ a 2 e − b ∣ c ∣ e − i c w d c for ( ( ∣ arg ( b ) ∣ ≤ 2 π ∧ arg ( w ) − 2 π < 2 π ) ∨ ( arg ( w ) − 2 π ≤ 2 π ∧ ∣ arg ( b ) ∣ < 2 π ) ∨ ( arg ( w ) − 2 π < 2 π ∧ ∣ arg ( b ) ∣ < 2 π ) ) ∧ ( ( ∣ arg ( b ) ∣ ≤ 2 π ∧ arg ( w ) + 2 π < 2 π ) ∨ ( arg ( w ) + 2 π ≤ 2 π ∧ ∣ arg ( b ) ∣ < 2 π ) ∨ ( arg ( w ) + 2 π < 2 π ∧ ∣ arg ( b ) ∣ < 2 π ) ) otherwise
=
/ 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 /
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< oo
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| | 2 -b*|c| -I*c*w
| | a *e *e dc otherwise
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\-oo
{ a 2 b ( 1 + i w b ) + a 2 b ( 1 − i w b ) for ( ( ∣ arg ( b ) ∣ ≤ π 2 ∧ ∣ arg ( w ) − π 2 ∣ < π 2 ) ∨ ( ∣ arg ( w ) − π 2 ∣ ≤ π 2 ∧ ∣ arg ( b ) ∣ < π 2 ) ∨ ( ∣ arg ( w ) − π 2 ∣ < π 2 ∧ ∣ arg ( b ) ∣ < π 2 ) ) ∧ ( ( ∣ arg ( b ) ∣ ≤ π 2 ∧ ∣ arg ( w ) + π 2 ∣ < π 2 ) ∨ ( ∣ arg ( w ) + π 2 ∣ ≤ π 2 ∧ ∣ arg ( b ) ∣ < π 2 ) ∨ ( ∣ arg ( w ) + π 2 ∣ < π 2 ∧ ∣ arg ( b ) ∣ < π 2 ) ) ∫ − ∞ ∞ a 2 e − b ∣ c ∣ e − i c w d c otherwise \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} ⎩ ⎨ ⎧ b ( 1 + b i w ) a 2 + b ( 1 − b i w ) a 2 − ∞ ∫ ∞ a 2 e − b ∣ c ∣ e − i c w d c for ( ( ∣ arg ( b ) ∣ ≤ 2 π ∧ arg ( w ) − 2 π < 2 π ) ∨ ( arg ( w ) − 2 π ≤ 2 π ∧ ∣ arg ( b ) ∣ < 2 π ) ∨ ( arg ( w ) − 2 π < 2 π ∧ ∣ arg ( b ) ∣ < 2 π ) ) ∧ ( ( ∣ arg ( b ) ∣ ≤ 2 π ∧ arg ( w ) + 2 π < 2 π ) ∨ ( arg ( w ) + 2 π ≤ 2 π ∧ ∣ arg ( b ) ∣ < 2 π ) ∨ ( arg ( w ) + 2 π < 2 π ∧ ∣ arg ( b ) ∣ < 2 π ) ) otherwise
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.
