La serie de Fourier es

\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})}

A continuación se determinarán los valores de cada coeficiente (a_0, a_n y  b_n).

Determinando el valor de «a_0»

Para conocer el valor de a_0, se agregará en cada miembro, la función \cos{m \omega_0 t}

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

Integrando en ambos miembros desde \-T/2 hasta T/2

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \int_{-T/2}^{T/2}{\frac{1}{2} a_0 \cdot \cos{m \omega_0 t} \, dt} + \int_{-T/2}^{T/2}{\sum_{n=1}^{\infty}{(a_n \cos{n \omega_0 t} \cdot \cos{m \omega_0 t} + b_n \sin{n \omega_0 t}\cdot \cos{m \omega_0 t})} \, dt}

Realizando acomodos adecuados en cada miembro

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \int_{-T/2}^{T/2}{ \left( \sum_{n=1}^{\infty}{a_n \cos{n \omega_0 t} \cdot \cos{m \omega_0 t}} + \sum_{n=1}^{\infty}{b_n \sin{n \omega_0 t}\cdot \cos{m \omega_0 t}} \right) \, dt}

\displaystyle = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \int_{-T/2}^{T/2}{ \left( \sum_{n=1}^{\infty}{a_n \cos{n \omega_0 t} \cdot \cos{m \omega_0 t}} \right) \, dt} + \int_{-T/2}^{T/2}{\left(\sum_{n=1}^{\infty}{b_n \sin{n \omega_0 t}\cdot \cos{m \omega_0 t}} \right) \, dt}

\displaystyle = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \sum_{n=1}^{\infty}{\int_{-T/2}^{T/2}{a_n \cos{n \omega_0 t} \cdot \cos{m \omega_0 t} } \, dt} + \sum_{n=1}^{\infty}{\int_{-T/2}^{T/2}{b_n \sin{n \omega_0 t}\cdot \cos{m \omega_0 t}} \, dt}

\displaystyle = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \sum_{n=1}^{\infty}{a_n \int_{-T/2}^{T/2}{\cos{n \omega_0 t} \cdot \cos{m \omega_0 t} \, dt}} + \sum_{n=1}^{\infty}{b_n \int_{-T/2}^{T/2}{\sin{n \omega_0 t} \cdot \cos{m \omega_0 t} \, dt}}

Si n \ne m, por propiedades de ortogonalidad

\displaystyle \int_{-T/2}^{T/2}{\cos{n \omega_0 t} \cdot \cos{m \omega_0 t} \, dt} = 0\displaystyle \int_{-T/2}^{T/2}{\sin{n \omega_0 t}\cdot \cos{m \omega_0 t} \, dt} = 0

Tendrá el siguiente resultado

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \sum_{n=1}^{\infty}{a_n \int_{-T/2}^{T/2}{\cos{n \omega_0 t} \cdot \cos{m \omega_0 t} \, dt}} + \sum_{n=1}^{\infty}{b_n \int_{-T/2}^{T/2}{\sin{n \omega_0 t}\cdot \cos{m \omega_0 t} \, dt}}

\displaystyle = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \sum_{n=1}^{\infty}{a_n \cdot 0} + \sum_{n=1}^{\infty}{b_n \cdot 0} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt}

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt}

Si m=0 en ambos miembros, resulta

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{(0 \cdot \omega_0 t)} \, dt} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{(0 \cdot \omega_0 t)} \, dt}

\displaystyle \int_{-T/2}^{T/2}{f(t) \, dt} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{dt}

Integrando, resulta lo siguiente

\displaystyle \int_{-T/2}^{T/2}{f(t) \, dt} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{dt} = \frac{1}{2} a_0 [t + C]_{-T/2}^{T/2}

\displaystyle \int_{-T/2}^{T/2}{f(t) \, dt} = \frac{1}{2} a_0 \left[ \frac{T}{2} - \left( -\frac{T}{2} \right) \right] = \frac{1}{2} a_0 \left( \frac{T}{2} + \frac{T}{2} \right)

\displaystyle \int_{-T/2}^{T/2}{f(t) \, dt} = \frac{1}{2} a_0 ( T ) = \frac{T}{2} a_0

Despejando a_0, finalmente, el resultado esperado es

\displaystyle \int_{-T/2}^{T/2}{f(t) \, dt} = \frac{T}{2} a_0

\displaystyle \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \, dt} = a_0

\displaystyle \therefore a_0 = \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \, dt}

Recordando también que

\displaystyle \frac{a_0}{2} = \frac{1}{T} \int_{-T/2}^{T/2}{f(t) \, dt}

Representa el valor promedio de f(t) durante un período.

Determinando el valor de «a_n»

Para obtener el valor de a_n, se agregará en cada miembro, la función \cos{m \omega_0 t}

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

Integrando en ambos miembros desde \-T/2 hasta T/2

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \int_{-T/2}^{T/2}{\frac{1}{2} a_0 \cdot \cos{m \omega_0 t} \, dt} + \int_{-T/2}^{T/2}{\sum_{n=1}^{\infty}{(a_n \cos{n \omega_0 t} \cdot \cos{m \omega_0 t} + b_n \sin{n \omega_0 t}\cdot \cos{m \omega_0 t})} \, dt}

Realizando acomodos adecuados en cada miembro

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \int_{-T/2}^{T/2}{ \left( \sum_{n=1}^{\infty}{a_n \cos{n \omega_0 t} \cdot \cos{m \omega_0 t}} + \sum_{n=1}^{\infty}{b_n \sin{n \omega_0 t}\cdot \cos{m \omega_0 t}} \right) \, dt}

\displaystyle = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \int_{-T/2}^{T/2}{ \left( \sum_{n=1}^{\infty}{a_n \cos{n \omega_0 t} \cdot \cos{m \omega_0 t}} \right) \, dt} + \int_{-T/2}^{T/2}{\left(\sum_{n=1}^{\infty}{b_n \sin{n \omega_0 t}\cdot \cos{m \omega_0 t}} \right) \, dt}

\displaystyle = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \sum_{n=1}^{\infty}{\int_{-T/2}^{T/2}{a_n \cos{n \omega_0 t} \cdot \cos{m \omega_0 t} } \, dt} + \sum_{n=1}^{\infty}{\int_{-T/2}^{T/2}{b_n \sin{n \omega_0 t}\cdot \cos{m \omega_0 t}} \, dt}

\displaystyle = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \sum_{n=1}^{\infty}{a_n \int_{-T/2}^{T/2}{\cos{n \omega_0 t} \cdot \cos{m \omega_0 t} \, dt}} + \sum_{n=1}^{\infty}{b_n \int_{-T/2}^{T/2}{\sin{n \omega_0 t} \cdot \cos{m \omega_0 t} \, dt}}

Si n = m \ne 0, por propiedades de ortogonalidad

\displaystyle \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} = 0
\displaystyle \int_{-T/2}^{T/2}{\cos{n \omega_0 t} \cdot \cos{m \omega_0 t} \, dt} = \frac{T}{2}
\displaystyle \int_{-T/2}^{T/2}{\sin{n \omega_0 t}\cdot \cos{m \omega_0 t} \, dt} = 0

Tendrá el siguiente resultado

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\cos{m \omega_0 t} \, dt} + \sum_{n=1}^{\infty}{a_n \int_{-T/2}^{T/2}{\cos{n \omega_0 t} \cdot \cos{m \omega_0 t} \, dt}} + \sum_{n=1}^{\infty}{b_n \int_{-T/2}^{T/2}{\sin{n \omega_0 t}\cdot \cos{m \omega_0 t} \, dt}}

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \frac{1}{2} a_0 \cdot 0 + \sum_{n=1}^{\infty}{a_n \cdot \frac{T}{2}} + \sum_{n=1}^{\infty}{b_n \cdot 0} = \sum_{n=1}^{\infty}{a_n \cdot \frac{T}{2}}

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \frac{T}{2} \sum_{n=1}^{\infty}{a_n} = \frac{T}{2} a_m

Despejando a_m

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = \frac{T}{2} a_m

\displaystyle \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt} = a_m

\displaystyle a_m = \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cdot \cos{m \omega_0 t} \, dt}

Finalmente, por la variable comodín (remplazando m por n), el valor de a_n es

\displaystyle \therefore a_n = \frac{2}{T} \int_{-T/2}^{T/2}{f(t) \cdot \cos{n \omega_0 t} \, dt}

donde n=0, \, 1, \, 2, \, \cdots.

Determinando el valor de «b_n»

Regresando a la serie de Fourier, si se multiplica en ambos miembros por la función \sin{m \omega_0 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 f(t) \cdot \sin{m \omega_0 t} = \frac{1}{2} a_0 \cdot \sin{m \omega_0 t} + \sum_{n=1}^{\infty}{(a_n \cos{n \omega_0 t} \cdot \sin{m \omega_0 t} + b_n \sin{n \omega_0 t} \cdot \sin{m \omega_0 t})}

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

Integrando en ambos miembros desde -T/2 hasta T/2

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \sin{m \omega_0 t} \, dt} = \int_{-T/2}^{T/2}{\frac{1}{2} a_0 \cdot \sin{m \omega_0 t} \, dt} + \int_{-T/2}^{T/2}{\sum_{n=1}^{\infty}{(a_n \cos{n \omega_0 t} \cdot \sin{m \omega_0 t)}} \, dt} + \int_{-T/2}^{T/2}{\sum_{n=1}^{\infty}{(b_n \sin{n \omega_0 t} \cdot \sin{m \omega_0 t})} \, dt}

Realizando acomodos adecuados

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \sin{m \omega_0 t} \, dt} = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\sin{m \omega_0 t} \, dt} +\sum_{n=1}^{\infty}{\int_{-T/2}^{T/2}{(a_n \cos{n \omega_0 t} \cdot \sin{m \omega_0 t)}} \, dt} + \sum_{n=1}^{\infty}{\int_{-T/2}^{T/2}{(b_n \sin{n \omega_0 t} \cdot \sin{m \omega_0 t})} \, dt}

\displaystyle = \frac{1}{2} a_0 \int_{-T/2}^{T/2}{\sin{m \omega_0 t} \, dt} +\sum_{n=1}^{\infty}{a_n \int_{-T/2}^{T/2}{\cos{n \omega_0 t} \cdot \sin{m \omega_0 t} \, dt}} + \sum_{n=1}^{\infty}{b_n \int_{-T/2}^{T/2}{\sin{n \omega_0 t} \cdot \sin{m \omega_0 t} \, dt}}

Si n = m \ne 0, por propiedades de ortogonalidad

\displaystyle \int_{-T/2}^{T/2}{\sin{m \omega_0 t} \, dt} = 0
\displaystyle \int_{-T/2}^{T/2}{\sin{n \omega_0 t} \cdot \cos{m \omega_0 t} \, dt} = 0
\displaystyle \int_{-T/2}^{T/2}{\sin{n \omega_0 t} \cdot \sin{m \omega_0 t} \, dt} = \frac{T}{2}

Y aplicándolos en la integrales, se tiene lo siguiente

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \sin{m \omega_0 t} \, dt} = \frac{1}{2} a_0 \cdot 0 +\sum_{n=1}^{\infty}{a_n \cdot 0} + \sum_{n=1}^{\infty}{b_n \int_{-T/2}^{T/2}{\sin{n \omega_0 t} \cdot \sin{m \omega_0 t} \, dt}}

\displaystyle \int_{-T/2}^{T/2}{f(t) \cdot \sin{m \omega_0 t} \, dt} = \sum_{n=1}^{\infty}{b_n \cdot \frac{T}{2}} = \frac{T}{2} \sum_{n=1}^{\infty}{b_n} = \frac{T}{2} b_m

Si se despeja b_m

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

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

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

Finalmente, por la variable comodín (remplazando m por n), el valor de b_n es

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

donde n=0, \, 1, \, 2, \, \cdots.

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Publicado por Cesar Reyes

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