Category:Definitions/Fourier Series
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This category contains definitions related to Fourier Series.
Related results can be found in Category: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 $\ds \int_\alpha^{\alpha + 2 \lambda} \map f x \rd x$ converges absolutely.
Let:
\(\ds a_n\) | \(=\) | \(\ds \dfrac 1 \lambda \int_\alpha^{\alpha + 2 \lambda} \map f x \cos \frac {n \pi x} \lambda \rd x\) | ||||||||||||
\(\ds b_n\) | \(=\) | \(\ds \dfrac 1 \lambda \int_\alpha^{\alpha + 2 \lambda} \map f x \sin \frac {n \pi x} \lambda \rd x\) |
Then:
- $\ds \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
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 $\ds \int_a^b \map f x \rd x$ converges absolutely.
Let:
\(\ds A_m\) | \(=\) | \(\ds \dfrac 2 {b - a} \int_a^b \map f x \cos \frac {2 m \pi \paren {x - a} } {b - a} \rd x\) | ||||||||||||
\(\ds B_m\) | \(=\) | \(\ds \dfrac 2 {b - a} \int_a^b \map f x \sin \frac {2 m \pi \paren {x - a} } {b - a} \rd x\) |
Then:
- $\ds \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$
is the Fourier Series for $f$.
Subcategories
This category has the following 2 subcategories, out of 2 total.
D
H
Pages in category "Definitions/Fourier Series"
The following 13 pages are in this category, out of 13 total.
F
- Definition:Fourier Coefficient
- Definition:Fourier Coefficient over Range 2 Pi
- Definition:Fourier Series
- Definition:Fourier Series over Range 2 Pi
- Definition:Fourier Series/Also defined as
- Definition:Fourier Series/Also known as
- Definition:Fourier Series/Formulation 1
- Definition:Fourier Series/Formulation 2
- Definition:Fourier Series/Fourier Coefficient
- Definition:Fourier Series/Fourier Coefficient/Range 2 Pi
- Definition:Fourier Series/Range 2 Pi