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

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.

$\mathrm{with}\left(\mathrm{inttrans}\right)\:$

$\mathrm{assume}\left(0<a\right)\&colon;$

$\mathrm{fourier}\left(\frac{3}{a^2+t^2}\&comma;t\&comma;w\right)$

$\frac{3\mathrm{\pi }\left(\mathrm{Heaviside}\left(-w\right){\&ExponentialE;}^{\mathrm{a~}w}+{\&ExponentialE;}^{-\mathrm{a~}w}\mathrm{Heaviside}\left(w\right)\right)}{\mathrm{a~}}$

$\mathrm{fourier}\left(\mathrm{diff}\left(f\left(x\right)\&comma;\mathrm{`\$`}\left(x\&comma;4\right)\right)\&comma;x\&comma;w\right)$

$w^4\mathrm{fourier}\left(f\left(x\right)\&comma;x\&comma;w\right)$

$F≔\mathrm{int}\left(g\left(x\right)h\left(t-x\right)\&comma;x=-\mathrm{\infty }..\mathrm{\infty }\right)\&colon;$

$\mathrm{fourier}\left(3F\&comma;t\&comma;w\right)$

$3\mathrm{fourier}\left(g\left(t\right)\&comma;t\&comma;w\right)\mathrm{fourier}\left(h\left(t\right)\&comma;t\&comma;w\right)$

$\mathrm{fourier}\left(t\exp \left(-3t\right)\mathrm{Heaviside}\left(t\right)\&comma;t\&comma;w\right)$

$\frac{1}{{\left(3+\mathrm{I}w\right)}^2}$

$\mathrm{fourier}\left(\frac{1}{{\left(4-\mathrm{I}t\right)}^{\frac{1}{3}}}\&comma;t\&comma;2+w\right)$

$\frac{\sqrt{3}\mathrm{\Gamma }\left(\frac{2}{3}\right){\&ExponentialE;}^{-8-4w}\mathrm{Heaviside}\left(2+w\right)}{{\left(2+w\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

$\mathrm{fourier}\left(\mathrm{diff}\left(y\left(t\right)\&comma;\mathrm{`\$`}\left(t\&comma;2\right)\right)-y\left(t\right)=\sin \left(at\right)\&comma;t\&comma;s\right)$

$-\left(s^2+1\right)\mathrm{fourier}\left(y\left(t\right)\&comma;t\&comma;s\right)=\mathrm{I}\mathrm{\pi }\left(-\mathrm{Dirac}\left(-s+\mathrm{a~}\right)+\mathrm{Dirac}\left(s+\mathrm{a~}\right)\right)$

$\mathrm{fourier}\left(\mathrm{BesselJ}\left(0\&comma;4{\left(t^2+1\right)}^{\frac{1}{2}}\right)\&comma;t\&comma;s\right)$

$\frac{8{\&ExponentialE;}^{\mathrm{I}s}\cos \left(s^2-16\right)\left(\mathrm{Heaviside}\left(s+4\right)-\mathrm{Heaviside}\left(s-4\right)\right)}{\sqrt{-s^2+16}}$

$\mathrm{addtable}\left(\mathrm{fourier}\&comma;\mathrm{myfunc}\left(t\right)\&comma;\frac{\mathrm{Myfunc}\left(s\right)}{1+s^2}\&comma;t\&comma;s\right)\&colon;$

$\mathrm{fourier}\left(\exp \left(3\mathrm{I}t\right)\mathrm{myfunc}\left(2t\right)\&comma;t\&comma;w\right)$

$\frac{2\mathrm{Myfunc}\left(\frac{w}{2}-\frac{3}{2}\right)}{w^2-6w+13}$

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