# Definition:Fourier Series/Formulation 1

## Definition

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 $f$.

### Fourier Coefficients

The constants:

- $a_0, a_1, a_2, \ldots, a_n, \ldots; b_1, b_2, \ldots, b_n, \ldots$

are the **Fourier coefficients** of $f$.

## Also see

- Derivation of Fourier Series over General Range, which provides the justification for this definition

- Results about
**Fourier series**can be found**here**.

## Source of Name

This entry was named for Joseph Fourier.

## Historical Note

Despite the fact that the Fourier series bears the name of Joseph Fourier, they were first studied by Leonhard Paul Euler.

Fourier himself made considerable use of this series during the course of his analysis of the behaviour of heat.

The first person to feel the need for a careful study of its convergence was Augustin Louis Cauchy.

In $1829$, Johann Peter Gustav Lejeune Dirichlet gave the first satisfactory proof about the sums of Fourier series for certain types of function.

The criteria set by Dirichlet were extended and generalized by Riemann in his $1854$ paper *Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe*.

## Sources

- 1961: I.N. Sneddon:
*Fourier Series*... (previous) ... (next): Chapter One: $\S 3$. Other Types of Whole-Range Series: $(1)$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 23$: Fourier Series: $23.1$, $23.2$