Editing Fourier transform (section)

=== Analytic integration of closed-form functions ===
Tables of [[closed-form expression|closed-form]] Fourier transforms, such as {{slink||Square-integrable functions, one-dimensional}} and {{slink|Discrete-time Fourier transform|Table of discrete-time Fourier transforms|nopage=y}}, are created by mathematically evaluating the Fourier analysis integral (or summation) into another closed-form function of frequency (<math>\xi</math> or <math>\omega</math>).<ref name="Zwillinger-2014">{{harvnb|Gradshteyn|Ryzhik|Geronimus|Tseytlin|2015}}</ref>  When mathematically possible, this provides a transform for a continuum of frequency values.

Many computer algebra systems such as [[Matlab]] and [[Mathematica]] that are capable of [[symbolic integration]] are capable of computing Fourier transforms analytically. For example, to compute the Fourier transform of {{math|1=cos(6π''t'') ''e''<sup>−π''t''<sup>2</sup></sup>}} one might enter the command {{code|integrate cos(6*pi*t) exp(−pi*t^2) exp(-i*2*pi*f*t) from -inf to inf}} into [[Wolfram Alpha]].<ref group=note>The direct command {{code|fourier transform of cos(6*pi*t) exp(−pi*t^2)}} would also work for Wolfram Alpha, although the options for the convention (see {{Section link|2=Other_conventions}}) must be changed away from the default option, which is actually equivalent to {{code|integrate cos(6*pi*t) exp(−pi*t^2) exp(i*omega*t) /sqrt(2*pi) from -inf to inf}}.</ref>