 ifourier - Maple Help

MTM

 ifourier
 inverse Fourier integral transform Calling Sequence ifourier(M) ifourier(M,u) ifourier(M,v, u) Parameters

 M - array or expression u - variable expr is transformed with respect to u v - parameter of transform Description

 • The ifourier function applies the inverse Fourier transform to M using the definition

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

 • The ifourier(M) calling sequence computes the element-wise inverse Fourier transform of M.  The result, R, is formed as R[i,j] = ifourier(M[i,j], v, u).
 • ifourier(F) is the inverse Fourier transform of the scalar F with default independent variable w.  If F is not a function of w, then F is  assumed to be a function of the independent variable returned by findsym(F,1). By default, the return value is a function of x.
 • If F = F(x), then ifourier returns a function of t. The integration above proceeds with respect to w.
 • ifourier(F,u) makes F a function of the variable u instead of the default x. The integration above proceeds with respect to w.
 • ifourier(F,v,u) takes F to be a function of v instead of the default w. The integration is then with respect to v. Examples

 > $\mathrm{with}\left(\mathrm{MTM}\right):$
 > $\mathrm{ifourier}\left(\frac{3}{1+{w}^{2}}\right)$
 $\frac{{3}{}{\mathrm{Heaviside}}{}\left({x}\right){}{{ⅇ}}^{{-}{x}}}{{2}}{+}\frac{{3}{}{{ⅇ}}^{{x}}{}{\mathrm{Heaviside}}{}\left({-}{x}\right)}{{2}}$ (1)
 > $\mathrm{ifourier}\left(\frac{3}{1+{x}^{2}}\right)$
 $\frac{{3}{}{\mathrm{Heaviside}}{}\left({t}\right){}{{ⅇ}}^{{-}{t}}}{{2}}{+}\frac{{3}{}{{ⅇ}}^{{t}}{}{\mathrm{Heaviside}}{}\left({-}{t}\right)}{{2}}$ (2)
 > $\mathrm{ifourier}\left(\frac{3}{1+{w}^{2}},s\right)$
 $\frac{{3}{}{\mathrm{Heaviside}}{}\left({s}\right){}{{ⅇ}}^{{-}{s}}}{{2}}{+}\frac{{3}{}{{ⅇ}}^{{s}}{}{\mathrm{Heaviside}}{}\left({-}{s}\right)}{{2}}$ (3)
 > $\mathrm{ifourier}\left(\frac{z\cdot 3}{1+{w}^{2}},z,t\right)$
 $\frac{{-}{3}{}{I}{}{\mathrm{Dirac}}{}\left({1}{,}{t}\right)}{{{w}}^{{2}}{+}{1}}$ (4)
 > $M≔\mathrm{Matrix}\left(\left[\frac{3}{1+{w}^{2}},\frac{z\cdot 3}{1+{w}^{2}}\right]\right):$
 > $\mathrm{ifourier}\left(M\right)$
 $\left[\begin{array}{cc}\frac{{3}{}{\mathrm{Heaviside}}{}\left({x}\right){}{{ⅇ}}^{{-}{x}}}{{2}}{+}\frac{{3}{}{{ⅇ}}^{{x}}{}{\mathrm{Heaviside}}{}\left({-}{x}\right)}{{2}}& \frac{{3}{}{z}{}\left({\mathrm{Heaviside}}{}\left({x}\right){}{{ⅇ}}^{{-}{x}}{+}{{ⅇ}}^{{x}}{}{\mathrm{Heaviside}}{}\left({-}{x}\right)\right)}{{2}}\end{array}\right]$ (5)