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inttrans

 invfourier
 inverse Fourier transform

 Calling Sequence invfourier(expr, w, t)

Parameters

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

Description

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

$f\left(t\right)=\frac{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F\left(w\right){ⅇ}^{Itw}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆw}{2\mathrm{\pi }}$

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{inttrans}\right):$
 > $\mathrm{invfourier}\left(\mathrm{fourier}\left(f\left(x\right),x,w\right),w,x-2\right)$
 ${f}{}\left({x}{-}{2}\right)$ (1)
 > $\mathrm{invfourier}\left(\frac{3}{1+{t}^{2}},t,w\right)$
 $\frac{{3}{}{\mathrm{Heaviside}}{}\left({w}\right){}{{ⅇ}}^{{-}{w}}}{{2}}{+}\frac{{3}{}{{ⅇ}}^{{w}}{}{\mathrm{Heaviside}}{}\left({-}{w}\right)}{{2}}$ (2)
 > $\mathrm{invfourier}\left(t\mathrm{exp}\left(-3t\right)\mathrm{Heaviside}\left(t\right),t,w\right)$
 $\frac{{1}}{{2}{}{\left({I}{}{w}{-}{3}\right)}^{{2}}{}{\mathrm{\pi }}}$ (3)
 > $\mathrm{invfourier}\left(\frac{1}{{\left(4-It\right)}^{\frac{1}{3}}},t,w\right)$
 $\frac{\sqrt{{3}}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right){}{{ⅇ}}^{{4}{}{w}}{}{\mathrm{Heaviside}}{}\left({-}{w}\right)}{{2}{}{\left({-}{w}\right)}^{{2}}{{3}}}{}{\mathrm{\pi }}}$ (4)
 > $\mathrm{addtable}\left(\mathrm{fourier},\mathrm{myfunc}\left(t\right),\mathrm{Myfunc}\left(s\right),t,s\right):$
 > $\mathrm{invfourier}\left(\mathrm{myfunc}\left(w\right),w,t\right)$
 $\frac{{\mathrm{Myfunc}}{}\left({-}{t}\right)}{{2}{}{\mathrm{\pi }}}$ (5)

Compatibility

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