3.8: Apéndice- Sech-Álgebra
- Page ID
- 84935
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La secante hiperbólica se define como
\[\text{sech} (x) = \dfrac{1}{\cosh (x)} \nonumber \]
\[\cosh^2 (x) - \sinh^2 (x) = 1 \nonumber \]
\[\text{sech}^2 (x) = 1 - \tanh^2 (x) \nonumber \]
\[\dfrac{d}{dx} \text{sech} (x) = -\tanh (x) \text{sech} (x) \nonumber \]
\[\dfrac{d^2}{dx^2} \text{sech} (x) = \text{sech} (x) [1 - 2\text{sech}^2 (x)] \nonumber \]
\[\int_{-\infty}^{+\infty} \text{sech} (x) dx = \pi \nonumber \]
| función\(f(t)\) | Transformada de Fourier\(\tilde{f} (\omega) = \int f(t) e^{-j\omega t} dt\) |
| \(\text{sech} (t)\) | \(\pi \text{sech} (\tfrac{\pi}{2} \omega)\) |
| \(\text{sech}^2 (t)\) | \(\tfrac{\pi \omega}{\sinh (\tfrac{\pi}{2} \omega)}\) |
| \(\text{sech}^3 (t)\) | \(\dfrac{1}{2} (1 + \omega^2) \pi \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(\text{sech}^5 (t)\) | \(\dfrac{1}{24} (\omega^4 + 10 \omega^2 + 9) \pi \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(\tanh (t) \text{sech} (t)\) | \(-j \pi \omega \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(\tanh^2 (t) \text{sech} (t)\) | \(\tfrac{1}{2} (1 - \omega^2) \pi \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(\tanh^3 (t) \text{sech} (t)\) | \(-j\tfrac{\omega}{6} (5 - \omega^2) \pi \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(\tanh (t) \text{sech}^3 (t)\) | \(-j\tfrac{\omega}{6} (1 + \omega^2) \pi \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(\tanh^2 (t) \text{sech}^3 (t)\) | \(\tfrac{1}{2} (1 + \omega^2) \pi \text{sech} (\dfrac{\pi}{2} \omega) - \tfrac{1}{24} (\omega^4 + 10\omega^2 + 9) \pi \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(t \tanh (t) \text{sech} (t)\) | \(\pi \text{sech} (\dfrac{\pi}{2} \omega) - \tfrac{\omega \pi^2}{2} \tanh (\dfrac{\pi}{2} \omega) \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(t\tanh^2 (t) \text{sech} (t)\) | \(-j \omega \pi \text{sech} (\dfrac{\pi}{2} \omega) - \tfrac{\pi^2}{4} (1 - \omega^2) \tanh (\dfrac{\pi}{2} \omega) \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(t\tanh^3 (t) \text{sech} (t)\) | \(\dfrac{1}{6} (5 - 3\omega^2) \pi \text{sech} (\dfrac{\pi}{2} \omega) - \tfrac{\omega \pi^2}{12} (5 -\omega^2) \tanh (\dfrac{\pi}{2} \omega) \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(t\tanh (t) \text{sech}^3 (t)\) | \(\dfrac{1}{6} (1 + 3\omega^2) \pi \text{sech} (\dfrac{\pi}{2} \omega) - \tfrac{\omega \pi^2}{12} (1 + \omega^2) \tanh (\dfrac{\pi}{2} \omega) \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(t \text{sech} (t)\) | \(-j \tfrac{\pi^2}{6} \tanh (\dfrac{\pi}{2} \omega) \text{sech} (\dfrac{\pi}{2} \omega)\) |
| \(t \text{sech}^3 (t)\) | \(-j\omega \pi \text{sech} (\dfrac{\pi}{2} \omega) - j \tfrac{\pi^2}{4} (1 + \omega^2) \tanh (\dfrac{\pi}{2} \omega) \text{sech} (\dfrac{\pi}{2} \omega)\) |