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

Expressions involving exponentials, polynomials, trigonometrics (sin, cos) and a variety of functions and other integral transforms can be transformed.

The invfourier function recognizes derivatives (diff or Diff) and integrals (int or Int).

Users can define the transforms of their own functions by using the function addtable.

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,invfourier) allows the use of the abbreviated form of this command.

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

$\mathrm{invfourier}\left(\mathrm{fourier}\left(f\left(x\right)\,x\,w\right)\,w\,x-2\right)$

$f\left(x-2\right)$

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

$\frac{3\mathrm{Heaviside}\left(w\right){\ⅇ}^{-w}}{2}+\frac{3{\ⅇ}^w\mathrm{Heaviside}\left(-w\right)}{2}$

$\mathrm{invfourier}\left(t\exp \left(-3t\right)\mathrm{Heaviside}\left(t\right)\,t\,w\right)$

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

$\mathrm{invfourier}\left(\frac{1}{{\left(4-\mathrm{I}t\right)}^{\frac{1}{3}}}\,t\,w\right)$

$\frac{\sqrt{3}\mathrm{\Gamma }\left(\frac{2}{3}\right){\ⅇ}^{4w}\mathrm{Heaviside}\left(-w\right)}{2{\left(-w\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\mathrm{\pi }}$

$\mathrm{addtable}\left(\mathrm{fourier}\,\mathrm{myfunc}\left(t\right)\,\mathrm{Myfunc}\left(s\right)\,t\,s\right)\:$

$\mathrm{invfourier}\left(\mathrm{myfunc}\left(w\right)\,w\,t\right)$

$\frac{\mathrm{Myfunc}\left(-t\right)}{2\mathrm{\pi }}$

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