Category:Examples of Fourier Series

From ProofWiki
Jump to navigation Jump to search

This category contains examples of Fourier Series.

Formulation 1

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

Let $\lambda \in \R_{>0}$ be a strictly positive real number.

Let $f: \R \to \R$ be a function such that $\displaystyle \int_\alpha^{\alpha + 2 \lambda} \map f x \rd x$ converges absolutely.


Let:

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


Then:

$\displaystyle \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$

is the Fourier Series for $f$.


Formulation 2

Let $a, b \in \R$ be real numbers.

Let $f: \R \to \R$ be a function such that $\displaystyle \int_a^b \map f x \rd x$ converges absolutely.


Let:

\(\displaystyle A_m\) \(=\) \(\displaystyle \dfrac 2 {b - a} \int_a^b \map f x \cos \frac {2 m \pi \paren {x - a} } {b - a} \rd x\)
\(\displaystyle B_m\) \(=\) \(\displaystyle \dfrac 2 {b - a} \int_a^b \map f x \sin \frac {2 m \pi \paren {x - a} } {b - a} \rd x\)


Then:

$\displaystyle \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$

is the Fourier Series for $f$.