Trigonometrie plana si sferica

Trigonometrie plana si sferica

Serii trigonometrice

Fie $$f:\mathbb{R}\rightarrow\mathbb{R}$$ o functie integrabila pe $$\mathbb{R}$$, periodica de perioada $$T=\frac{2\pi}{\omega}$$ care poate fi reprezentata printr-o serie trigonometrica
@df(x)=\frac{a_0}{2}+\sum_{n=1}^\infty\left(a_n\cos n\omega x+b_n\sin n\omega x\right).@d

Atunci coeficientii $$a_0,a_n,b_n$$ sunt dati de formulele
@d\begin{array}{l}
  a_0=\displaystyle\frac{2}{T}\int\limits_\alpha^{\alpha+T}f(x)dx\\
  a_n=\displaystyle\frac{2}{T}\int\limits_\alpha^{\alpha+T}f(x)\cos n\omega xdx,\ n\ge1\\
  b_n=\displaystyle\frac{2}{T}\int\limits_\alpha^{\alpha+T}f(x)\sin n\omega xdx,\ n\ge1
\end{array}
@d

Integralele nu depind de $$\alpha$$ si de obicei se alege $$\alpha=0$$ sau $$\alpha=-\frac{T}{2}$$.

Pentru valorile lui $$\alpha$$ anterioare, daca notam $$T=2l\Rightarrow\omega=\frac{2\pi}{T}=\frac{\pi}{l}$$, formulele anterioare devin:

$$
\begin{array}{l}
  a_0=\displaystyle\frac{1}{l}\int\limits_0^{2l}f(x)dx\\
  a_n=\displaystyle\frac{1}{l}\int\limits_0^{2l}f(x)\cos \frac{n\pi x}{l}dx,\ n\ge1\\
  b_n=\displaystyle\frac{1}{l}\int\limits_0^{2l}f(x)\sin \frac{n\pi x}{l}dx,\ n\ge1
\end{array}
$$         sau         $$
\begin{array}{l}
  a_0=\displaystyle\frac{1}{l}\int\limits_{-l}^{l}f(x)dx \\
  a_n=\displaystyle\frac{1}{l}\int\limits_{-l}^{l}f(x)\cos \frac{n\pi x}{l}dx,\ n\ge1 \\
  b_n=\displaystyle\frac{1}{l}\int\limits_{-l}^{l}f(x)\sin \frac{n\pi x}{l}dx,\ n\ge1
\end{array}
$$

Formulele anterioare se numesc formulele Euler-Fourier, iar seria corespunzatoare se numeste serie Fourier trigonometrica asociata functiei $$f$$.

Daca functia $$f$$ periodica de perioada $$T=2l$$ este para, coeficientii Fourier sunt

\begin{equation}\label{ec27}
\begin{array}{l}
  a_0=\frac{2}{l}\int\limits_{0}^{l}f(x)dx \\
  a_n=\frac{2}{l}\int\limits_{0}^{l}f(x)\cos \frac{n\pi x}{l}dx,\ n\ge1 \\
  b_n=0,\ n\ge1
\end{array}
\end{equation}iar seria Fourier trigonometrica este numai de cosinusuri:
\begin{equation*}f(x)=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos \frac{n\pi x}{l}.\end{equation*}
Daca functia $$f$$ periodica de perioada $$T=2l$$ este impara, coeficientii Fourier sunt

@d
\begin{array}{l}
  a_0=0 \\
  a_n=0,\ n\ge1 \\
  b_n=\frac{2}{l}\int\limits_{0}^{l}f(x)\sin \frac{n\pi x}{l}dx,\ n\ge1
\end{array}
@d

iar seria Fourier trigonometrica este numai de sinusuri:
\begin{equation*}f(x)=\sum_{n=1}^\infty b_n\sin \frac{n\pi x}{l}.\end{equation*}

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