# 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 $f$.

## Pages in category "Examples of Fourier Series"

The following 18 pages are in this category, out of 18 total.

### F

- Fourier Series/1 over -1 to 0, Cosine of x over 0 to 1
- Fourier Series/4 minus x squared over Range of 2
- Fourier Series/Absolute Value of x over Minus Pi to Pi
- Fourier Series/Cosine of Non-Integer Multiple of x over 0 to Pi
- Fourier Series/Cosine of x over Minus Pi to Zero, Minus Cosine of x over Zero to Pi
- Fourier Series/Cosine Series for x over 0 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/Half-Range Sine Series of Cosine over 0 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/Pi Squared minus x Squared over Minus Pi to Pi
- Fourier Series/Sine of Half x over 0 to Pi, Minus Sine of Half x over Pi to 2 Pi
- Fourier Series/Square of x minus pi, Square of pi
- Fourier Series/x by Pi minus x over 0 to Pi
- Fourier Series/x over 0 to 2, x-2 over 2 to 4
- Fourier Series/x over Minus Pi to Pi
- Fourier Series/x squared over Minus Pi to Pi