fourier - Maple Help

inttrans

 fourier
 Fourier transform

 Calling Sequence fourier(expr, t, w)

Parameters

 expr - expression, equation, or set of equations and/or expressions to be transformed t - variable expr is transformed with respect to t w - parameter of transform opt - option to run this under (optional)

Description

 • The fourier function computes the Fourier transform (F(w)) of expr (f(t)) with respect to t, using the definition

$F\left(w\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f\left(t\right){ⅇ}^{-Iwt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆt$

 • Expressions involving complex exponentials, polynomials, trigonometrics (sin, cos) and a variety of functions and other integral transforms can be transformed.
 • The fourier function recognizes derivatives (diff or Diff) and integrals (int or Int).
 • Users can add their own functions to fourier's internal lookup table with the function inttrans[addtable].
 • fourier  recognizes the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).
 • The program first attempts to classify the function simply, from the lookup table.  Then it considers various cases, including a piecewise decomposition, products, powers, sums, and rational polynomials.  Finally, if all other methods fail, the program will resort to integration.  If the option opt is set to 'NO_INT', then the program will not integrate. This will increase the speed at which the transform will run.
 • The command with(inttrans,fourier) allows the use of the abbreviated form of this command.
 • For information on computing Fourier transforms on signal data, see Fourier Transforms in Maple.

Examples

 > $\mathrm{with}\left(\mathrm{inttrans}\right):$
 > $\mathrm{assume}\left(0
 > $\mathrm{fourier}\left(\frac{3}{{a}^{2}+{t}^{2}},t,w\right)$
 $\frac{{3}{}{\mathrm{\pi }}{}\left({\mathrm{Heaviside}}{}\left({-}{w}\right){}{{ⅇ}}^{{\mathrm{a~}}{}{w}}{+}{{ⅇ}}^{{-}{\mathrm{a~}}{}{w}}{}{\mathrm{Heaviside}}{}\left({w}\right)\right)}{{\mathrm{a~}}}$ (1)
 > $\mathrm{fourier}\left(\frac{{ⅆ}^{4}}{ⅆ{x}^{4}}f\left(x\right),x,w\right)$
 ${{w}}^{{4}}{}{\mathrm{fourier}}{}\left({f}{}\left({x}\right){,}{x}{,}{w}\right)$ (2)
 > $F≔{∫}_{-\mathrm{∞}}^{\mathrm{∞}}g\left(x\right)h\left(t-x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx:$
 > $\mathrm{fourier}\left(3F,t,w\right)$
 ${3}{}{\mathrm{fourier}}{}\left({g}{}\left({t}\right){,}{t}{,}{w}\right){}{\mathrm{fourier}}{}\left({h}{}\left({t}\right){,}{t}{,}{w}\right)$ (3)
 > $\mathrm{fourier}\left(t{ⅇ}^{-3t}\mathrm{Heaviside}\left(t\right),t,w\right)$
 $\frac{{1}}{{\left({3}{+}{I}{}{w}\right)}^{{2}}}$ (4)
 > $\mathrm{fourier}\left(\frac{1}{{\left(4-It\right)}^{\frac{1}{3}}},t,2+w\right)$
 $\frac{\sqrt{{3}}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right){}{{ⅇ}}^{{-}{8}{-}{4}{}{w}}{}{\mathrm{Heaviside}}{}\left({2}{+}{w}\right)}{{\left({2}{+}{w}\right)}^{{2}}{{3}}}}$ (5)
 > $\mathrm{fourier}\left(\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}y\left(t\right)-y\left(t\right)=\mathrm{sin}\left(at\right),t,s\right)$
 ${-}\left({{s}}^{{2}}{+}{1}\right){}{\mathrm{fourier}}{}\left({y}{}\left({t}\right){,}{t}{,}{s}\right){=}{I}{}{\mathrm{\pi }}{}\left({-}{\mathrm{Dirac}}{}\left({-}{s}{+}{\mathrm{a~}}\right){+}{\mathrm{Dirac}}{}\left({s}{+}{\mathrm{a~}}\right)\right)$ (6)
 > $\mathrm{fourier}\left(\mathrm{BesselJ}\left(0,4{\left({t}^{2}+1\right)}^{\frac{1}{2}}\right),t,s\right)$
 $\frac{{8}{}{{ⅇ}}^{{I}{}{s}}{}{\mathrm{cos}}{}\left({{s}}^{{2}}{-}{16}\right){}\left({\mathrm{Heaviside}}{}\left({s}{+}{4}\right){-}{\mathrm{Heaviside}}{}\left({s}{-}{4}\right)\right)}{\sqrt{{-}{{s}}^{{2}}{+}{16}}}$ (7)
 > $\mathrm{addtable}\left(\mathrm{fourier},\mathrm{myfunc}\left(t\right),\frac{\mathrm{Myfunc}\left(s\right)}{1+{s}^{2}},t,s\right):$
 > $\mathrm{fourier}\left({ⅇ}^{3It}\mathrm{myfunc}\left(2t\right),t,w\right)$
 $\frac{{2}{}{\mathrm{Myfunc}}{}\left(\frac{{w}}{{2}}{-}\frac{{3}}{{2}}\right)}{{{w}}^{{2}}{-}{6}{}{w}{+}{13}}$ (8)

Compatibility

 • The inttrans[fourier] command was updated in Maple 2019.