Definition:Trigonometric Series

Definition

A trigonometric series is a series of the type:

$\map S x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$

where:

the domain of $x$ is the set of real numbers $\R$

the coefficients $a_0, a_1, a_2, \ldots, a_n, \ldots; b_1, b_2, \ldots, b_n, \ldots$ are real numbers independent of $x$.

The coefficient $a_0$ has the factor $\dfrac 1 2$ applied for convenience of algebraic manipulation.

Complex Form

A trigonometric series can be expressed in a form using complex functions as follows:

$\map S x = \ds \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$

where:

the domain of $x$ is the set of real numbers $\R$

the coefficients $\ldots, c_{-n}, \ldots, c_{-2}, c_{-1}, c_0, c_1, c_2, \ldots, c_n, \ldots$ are real numbers independent of $x$

$c_{-n} = \overline {c_n}$ where $\overline {c_n}$ is the complex conjugate of $c_n$.

Also known as

Some sources give this as trigonometrical series. $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to standardise on the shorter version.

Sources

• 1961: I.N. Sneddon: Fourier Series ... (next): Chapter One: $\S 1$. Trigonometrical Series: $(1)$