Definition:Trigonometric Series/Complex Form

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Definition

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

$S \left({x}\right) = \displaystyle \sum_{n \mathop = -\infty}^\infty c_n e^{i n x}$

where:

the domain of $x$ is the set of real numbers $\R$
the coefficients $\cdots c_{-n}, \ldots, c_{-2}, c_{-1}, c_0, c_1, c_2, \ldots, c_n, \ldots\ldots$ are real numbers independent of $x$
$c_{- n} = \overline {c_n}$ where $\overline {c_n}$ is the complex conjugate of $c_n$.


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