blog, fourier

Diferenciación de la serie de Fourier. Fourier.

Se observa que la diferenciación término por término de una serie trigonométrica

\displaystyle \frac{1}{2} a_0 + \sum_{n=1}^{\infty}{a_n \cos{\omega_0 t} + b_n \sin{n\omega_0 t}}

multiplica los coeficientes a_n y b_n por \pm n omega_0. Donde la diferenciación tiende a disminuir la convergencia y puede resultar en divergencia.

Demostración del teorema de diferenciación de la serie de Fourier

Teorema. Si f(t) es continua cuando -T/2 < t < T/2 con f(-T/2) = f(T/2), y si la derivada f'(t) es continua por tramos, y diferenciable, entonces la serie de Fourier

\displaystyle f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}{(a_n \cos{n \omega_0 t} + b_n \sin{n \omega_0 t})}

se puede diferenciar término por término para obtener

\displaystyle f'(t) = \sum_{n=1}^{k}{n\omega_0 (-a_n \sin{n\omega_0 t} + b_n \cos{n \omega t})}

Demostración.

Partiendo de la serie de Fourier, se deriva la función con respecto a «t«

\displaystyle f(t) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty}{(a_n \cos{n \omega_0 t} + b_n \sin{n \omega_0 t})}

\displaystyle \frac{d}{dt} f(t) = \frac{1}{2} \frac{d}{dt}(a_0) + \frac{d}{dt} \left[\sum_{n=1}^{\infty}{(a_n \cos{n \omega_0 t} + b_n \sin{n \omega_0 t})} \right]

\displaystyle \frac{d}{dt} f(t) = \frac{1}{2} \frac{d}{dt}\left[\frac{2}{T}\int_{-T/2}^{T/2}{f(t) \, dt} \right] + \frac{d}{dt} \left\{\sum_{n=1}^{\infty}{\left[\left(\frac{2}{T}\int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \cdot \cos{n \omega_0 t} + \left(\frac{2}{T}\int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) \cdot \sin{n \omega_0 t} \right]} \right\}

Atendiendo sólo el segundo término de la derivada del miembro derecho

\displaystyle \frac{d}{dt} \left\{\sum_{n=1}^{\infty}{\left[\left(\frac{2}{T}\int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \cdot \cos{n \omega_0 t} + \left(\frac{2}{T}\int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) \cdot \sin{n \omega_0 t} \right]} \right\}

Se deriva lo siguiente

\displaystyle \sum_{n=1}^{\infty}{\left[\frac{d}{dt} \left[ \left( \frac{2}{T}\int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \cdot \cos{n \omega_0 t} \right] + \frac{d}{dt} \left[\left(\frac{2}{T}\int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) \cdot \sin{n \omega_0 t} \right] \right]}

Derivando el primer término de la suma

\displaystyle \frac{d}{dt} \left[ \frac{2}{T} \left( \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \cdot \cos{n \omega_0 t} \right]

\displaystyle = \left( \frac{2}{T} \int_{-T/2}^{T/2}{\left(\frac{d}{dt} [f(t)] \cdot \cos{n \omega_0 t} + f(t) (-n\omega_0 \sin{n \omega_0 t}) \right) \, dt} \right) \cdot \cos{n \omega_0 t} + \left( \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \cdot (-n \omega_0 \sin{n \omega_0 t})

\displaystyle = \left(\frac{2}{T} \int_{-T/2}^{T/2}{\left(f'(t) \cos{n \omega_0 t} - n \omega_0 f(t) \sin{n \omega_0 t} \right) \, dt} \right) \cos{n \omega_0 t} - n \omega_0 \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \sin{n \omega_0 t}

\displaystyle = \left( \frac{2}{T}\int_{-T/2}^{T/2}{f'(t) \cos{n \omega_0 t} \, dt} - n \omega_0 \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n \omega_0 t} \, dt} \right) \cos{n \omega_0 t} - n \omega_0 \left( \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \sin{n \omega_0 t}

\displaystyle = \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \cos{n \omega_0 t} \, dt} \right) \cos{n \omega_0 t} - n \omega_0 \left(\frac{2}{T}\int_{-T/2}^{T/2}{f(t) \sin{n \omega_0 t} \, dt} \right) \cos{n \omega_0 t} - n \omega_0 \left( \frac{2}{T}\int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \sin{n \omega_0 t}

Derivando el segundo término de la suma

\displaystyle \frac{d}{dt} \left[\left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) \cdot \sin{n \omega_0 t} \right]

\displaystyle = \left(\frac{2}{T} \int_{-T/2}^{T/2}{\left(\frac{d}{dt} [f(t)] \cdot \sin{n\omega_0 t} + f(t) (n\omega_0 \cos{n\omega_0 t}) \right) \, dt} \right) \cdot \sin{n \omega_0 t} + \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) \cdot (n \omega_0 \cos{n \omega_0 t})

\displaystyle = \left(\frac{2}{T} \int_{-T/2}^{T/2}{\left(f'(t) \sin{n\omega_0 t} + n\omega_0 \cdot f(t) \cos{n\omega_0 t} \right) \, dt} \right) \sin{n \omega_0 t} + \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) (n \omega_0 \cos{n \omega_0 t})

\displaystyle = \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \sin{n\omega_0 t} \, dt}+ n\omega_0 \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n\omega_0 t} \, dt} \right) \sin{n \omega_0 t} + \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) (n \omega_0 \cos{n \omega_0 t})

\displaystyle = \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \sin{n\omega_0 t} \, dt} \right) \sin{n \omega_0 t} + n\omega_0 \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n\omega_0 t} \, dt} \right) \sin{n \omega_0 t} + n\omega_0 \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) \cos{n \omega_0 t}

Regresado a la suma

\displaystyle \sum_{n=1}^{\infty}{\left[\frac{d}{dt} \left[ \left( \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \cdot \cos{n \omega_0 t} \right] + \frac{d}{dt} \left[\left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) \cdot \sin{n \omega_0 t} \right] \right]}

\displaystyle = \sum_{n=1}^{\infty}{\left[ \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \cos{n \omega_0 t} \, dt} \right) \cos{n \omega_0 t} - n \omega_0 \left(\frac{2}{T}\int_{-T/2}^{T/2}{f(t) \sin{n \omega_0 t} \, dt} \right) \cos{n \omega_0 t} - n \omega_0 \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \sin{n \omega_0 t} + \right.}

\displaystyle \left. + \left(\frac{2}{T}\int_{-T/2}^{T/2}{f'(t) \sin{n\omega_0 t} \, dt} \right) \sin{n \omega_0 t} + n\omega_0 \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n\omega_0 t} \, dt} \right) \sin{n \omega_0 t} + n\omega_0 \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) \cos{n \omega_0 t} \right]

\displaystyle = \sum_{n=1}^{\infty}{\left[ \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \cos{n \omega_0 t} \, dt} \right) \cos{n \omega_0 t} + \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \sin{n\omega_0 t} \, dt} \right) \sin{n \omega_0 t} \right]}

Regresando a la primera derivada de la serie de Fourier y sustituyendo los resultados obtenidos

\displaystyle \frac{d}{dt} f(t) = \frac{1}{2} \frac{d}{dt}\left[\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \, dt} \right] + \frac{d}{dt} \left\{\sum_{n=1}^{\infty}{\left[\left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt} \right) \cdot \cos{n \omega_0 t} + \left(\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n\omega_0 t} \, dt} \right) \cdot \sin{n \omega_0 t} \right]} \right\}

\displaystyle f'(t) = \frac{1}{2} \frac{d}{dt}\left[\frac{2}{T} \int_{-T/2}^{T/2}{f(t) \, dt} \right] + \sum_{n=1}^{\infty}{\left[ \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \cos{n \omega_0 t} \, dt} \right) \cos{n \omega_0 t} + \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \sin{n\omega_0 t} \, dt} \right) \sin{n \omega_0 t} \right]}

\displaystyle f'(t) = \frac{1}{2} \cdot \frac{2}{T} \int_{-T/2}^{T/2}{\frac{d}{dt}[f(t)] \, dt} + \sum_{n=1}^{\infty}{\left[ \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \cos{n \omega_0 t} \, dt} \right) \cos{n \omega_0 t} + \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \sin{n\omega_0 t} \, dt} \right) \sin{n \omega_0 t} \right]}

\displaystyle f'(t) = \frac{1}{2} \cdot \frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \, dt} + \sum_{n=1}^{\infty}{\left[ \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \cos{n \omega_0 t} \, dt} \right) \cos{n \omega_0 t} + \left(\frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \sin{n\omega_0 t} \, dt} \right) \sin{n \omega_0 t} \right]}

\displaystyle f'(t) = \frac{1}{2} \alpha_0 + \sum_{n=1}^{\infty}{(\alpha_n \cos{n \omega_0 t} + \beta_n \sin{n \omega_0 t})}

Donde \displaystyle \alpha_0 = \frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \, dt}, \displaystyle \alpha_n = \frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \cos{n \omega_0 t} \, dt} y \displaystyle \beta_n = \frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \sin{n\omega_0 t} \, dt}.

De \alpha_0, tiene una siguiente expresión (utilizando integración directa)

\displaystyle \alpha_0 = \frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \, dt}

\displaystyle \alpha_0 = \frac{2}{T} \left[f(t) + C \right]_{-T/2}^{T/2}

\displaystyle \alpha_0 = \frac{2}{T} \left[f \left(\frac{T}{2}\right) - f \left(-\frac{T}{2} \right) \right]

Si \displaystyle f \left( \frac{T}{2} \right) = f \left( - \frac{T}{2} \right), entonces

\displaystyle \alpha_0 = 0

De \alpha_n, tiene una siguiente expresión (utilizando integración por partes)

\displaystyle \alpha_n = \frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \cos{n \omega_0 t} \, dt}

\displaystyle \alpha_n = \frac{2}{T} \left\{ \left[f(t) \cos{n \omega_0 t} + C\right]_{-T/2}^{T/2} - \int_{-T/2}^{T/2}{f(t) (-n\omega_0 \sin{n \omega_0 t}) \, dt} \right\}

\displaystyle \alpha_n = \frac{2}{T} \left[f(t) \cos{n \omega_0 t} + C\right]_{-T/2}^{T/2} - \frac{2}{T} \int_{-T/2}^{T/2}{f(t) (-n\omega_0 \sin{n \omega_0 t}) \, dt}

\displaystyle \alpha_n = \frac{2}{T} \left[f \left(\frac{T}{2} \right) \cos{n \omega_0 (\frac{T}{2})} - f \left(-\frac{T}{2} \right) \cos{n \omega_0 (-\frac{T}{2})} \right] + n \omega_0 \cdot \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n \omega_0 t} \, dt}

\displaystyle \alpha_n = \frac{2}{T} \left[f \left(\frac{T}{2} \right) \cos{n \omega_0 (\frac{T}{2})} - f \left(\frac{T}{2} \right) \cos{n \omega_0 (\frac{T}{2})} \right] + n \omega_0 \cdot \frac{2}{T}\int_{-T/2}^{T/2}{f(t) \sin{n \omega_0 t} \, dt}

\displaystyle \alpha_n = n \omega_0 \cdot \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \sin{n \omega_0 t} \, dt} = n \omega_0 \cdot b_n

\displaystyle \alpha_n = n \omega_0 b_n

Recordando que \displaystyle f \left( \frac{T}{2} \right) = f \left( - \frac{T}{2} \right).

De \beta_n, tiene una siguiente expresión (utilizando integración por partes)

\displaystyle \beta_n = \frac{2}{T} \int_{-T/2}^{T/2}{f'(t) \sin{n\omega_0 t} \, dt}

\displaystyle \beta_n = \frac{2}{T} \left[f(t) \sin{n \omega_0 t} \right]_{-T/2}^{T/2} - \frac{2}{T} \int_{-T/2}^{T/2}{f(t) (n\omega_0 \cos{n \omega_0 t}) \, dt}

\displaystyle \beta_n = \frac{2}{T} \left[f \left( \frac{T}{2} \right) \sin{n \omega_0 (\frac{T}{2})} - f \left(-\frac{T}{2} \right) \sin{n \omega_0 (-\frac{T}{2})} \right] - n\omega_0 \cdot \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt}

\displaystyle \beta_n = \frac{2}{T} \left[f \left( \frac{T}{2} \right) \sin{n \omega_0 (\frac{T}{2})} + f \left(\frac{T}{2} \right) \sin{n \omega_0 (\frac{T}{2})} \right] - n\omega_0 \cdot \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt}

\displaystyle \beta_n = \frac{2}{T} \left[2 f \left( \frac{T}{2} \right) \sin{n \omega_0 (\frac{T}{2})} \right] - n\omega_0 \cdot \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt}

\displaystyle \beta_n = \frac{4}{T} f \left( \frac{T}{2} \right) \sin{n \omega_0 (\frac{T}{2})} - n\omega_0 \cdot \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cos{n \omega_0 t} \, dt}

\displaystyle \beta_n = \frac{4}{T} f \left( \frac{T}{2} \right) \sin{n \omega_0 (\frac{T}{2})} - n\omega_0 a_n

Recordando que \displaystyle f \left( \frac{T}{2} \right) = f \left( - \frac{T}{2} \right).

Regresando con la continuación de la derivada de f'(t) y sustituyendo

\displaystyle f'(t) = \frac{1}{2} \alpha_0 + \sum_{n=1}^{\infty}{(\alpha_n \cos{n \omega_0 t} + \beta_n \sin{n \omega_0 t})}

\displaystyle f'(t) = \frac{1}{2} (0) + \sum_{n=1}^{\infty}{\left[n \omega_0 b_n \cos{n \omega_0 t} + \left[ \frac{4}{T} f \left( \frac{T}{2} \right) \sin{n \omega_0 (\frac{T}{2})} - n\omega_0 a_n \right] \sin{n \omega_0 t}\right]}

\displaystyle f'(t) = \sum_{n=1}^{\infty}{n \omega_0 b_n \cos{n \omega_0 t}} + \sum_{n=1}^{\infty}{ \frac{4}{T}f \left( \frac{T}{2} \right) \sin{n \omega_0 (\frac{T}{2})} \sin{n\omega_0 t}} - \sum_{n=1}^{\infty}{n\omega_0 a_n \sin{n \omega_0 t}}

\displaystyle f'(t) = \sum_{n=1}^{\infty}{n \omega_0 b_n \cos{n \omega_0 t}} + \frac{4}{T}f \left( \frac{T}{2} \right) \sum_{n=1}^{\infty}{\sin{n \omega_0 (\frac{T}{2})} \sin{n\omega_0 t}} - \sum_{n=1}^{\infty}{n\omega_0 a_n \sin{n \omega_0 t}}

\displaystyle f'(t) = \sum_{n=1}^{\infty}{n \omega_0 b_n \cos{n \omega_0 t}} + \frac{4}{T}f \left( \frac{T}{2} \right) \sum_{n=1}^{\infty}{\sin{n \pi} \sin{n\omega_0 t}} - \sum_{n=1}^{\infty}{n\omega_0 a_n \sin{n \omega_0 t}}

\displaystyle f'(t) = \sum_{n=1}^{\infty}{n \omega_0 b_n \cos{n \omega_0 t}} - \sum_{n=1}^{\infty}{n\omega_0 a_n \sin{n \omega_0 t}}

\displaystyle f'(t) = \sum_{n=1}^{\infty}{(n \omega_0 b_n \cos{n \omega_0 t} - n\omega_0 a_n \sin{n \omega_0 t})}

\displaystyle f'(t) = \sum_{n=1}^{\infty}{(n \omega_0)(b_n \cos{n \omega_0 t} - a_n \sin{n \omega_0 t})}

\displaystyle \therefore f'(t) = \sum_{n=1}^{\infty}{n \omega_0(-a_n \sin{n \omega_0 t} + b_n \cos{n \omega_0 t})}

Y el teorema queda demostrado.


Deja una respuesta

Introduce tus datos o haz clic en un icono para iniciar sesión:

Logo de WordPress.com

Estás comentando usando tu cuenta de WordPress.com. Salir /  Cambiar )

Imagen de Twitter

Estás comentando usando tu cuenta de Twitter. Salir /  Cambiar )

Foto de Facebook

Estás comentando usando tu cuenta de Facebook. Salir /  Cambiar )

Conectando a %s

Este sitio usa Akismet para reducir el spam. Aprende cómo se procesan los datos de tus comentarios.