Category:Examples of Fourier Series
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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 $\ds \int_\alpha^{\alpha + 2 \lambda} \map f x \rd x$ 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 6 subcategories, out of 6 total.
E
F
Pages in category "Examples of Fourier Series"
The following 12 pages are in this category, out of 12 total.
F
- Fourier Series for Logarithm of Sine of x over 0 to Pi
- Fourier Series/1 over -1 to 0, Cosine of Pi x over 0 to 1
- Fourier Series/4 minus x squared over Range of 2
- Fourier Series/Cosine of x over Minus Pi to Zero, Minus Cosine of x over Zero to Pi
- Fourier Series/Exponential of x over Minus Pi to Pi
- Fourier Series/Fourth Power of x over Minus Pi to Pi
- Fourier Series/Minus Pi over 0 to Pi, x minus Pi over Pi to 2 Pi
- Fourier Series/Pi minus x over 0 to 2 Pi
- Fourier Series/Sixth Power of x over Minus Pi to Pi
- Fourier Series/Square of x minus pi, Square of pi
- Fourier Series/x over 0 to 2, x-2 over 2 to 4
- Fourier Series/x squared over Minus Pi to Pi