Category:Examples of Fourier Series

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This category contains examples of Fourier Series.

Let $\alpha \in \R$ be a real number.

Let $f: \R \to \R$ be a function such that $\displaystyle \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \rd x$ converges absolutely.


Let:

\(\displaystyle a_n\) \(=\) \(\displaystyle \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \cos n x \rd x\)
\(\displaystyle b_n\) \(=\) \(\displaystyle \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \sin n x \rd x\)


Then:

$\displaystyle \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \left({a_n \cos n x + b_n \sin n x}\right)$

is called the Fourier Series for $f$.