Definition:Trigonometric Series/Complex Form

Definition

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 see

• Equivalence of Definitions of Trigonometric Series

Sources

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