ifourier (Symbolic Math Toolbox)

Inverse Fourier integral transform

Syntax

f = ifourier(F)
f = ifourier(F,u)
f = ifourier(F,v,u)

Description

f = ifourier(F) is the inverse Fourier transform of the scalar symbolic object F with default independent variable w. The default return is a function of x. The inverse Fourier transform is applied to a function of w and returns a function of x.

If F = F(x), ifourier returns a function of t.

By definition

f = ifourier(F,u) makes f a function of u instead of the default x.

Here u is a scalar symbolic object.

f = ifourier(F,v,u) takes F to be a function of v and f to be a function of u instead of the default w and x, respectively.

Examples

Inverse Fourier Transform
MATLAB Command

syms a real
f = exp(-w^2/(4*a^2))
F = ifourier(f)
F = simple(F)
returns
a*exp(-x^2*a^2)/pi^(1/2)

g = exp(-abs(x))

ifourier(g)
returns
1/(1+t^2)/pi

f = 2*exp(-abs(w)) - 1

simple(ifourier(f,t))
returns
-(-2+pi*Dirac(t))/(1+t^2)/pi

syms w real
f = exp(-w^2*abs(v))*sin(v)/v

ifourier(f,v,t)
returns
-1/2*(-atan((t+1)/w^2) +atan((-1+t)/w^2))/pi

Inverse Fourier Transform	MATLAB Command
	`syms a real` `f = exp(-w^2/(4a^2))` `F = ifourier(f)` `F = simple(F)` returns `aexp(-x^2*a^2)/pi^(1/2)`
	`g = exp(-abs(x))` `ifourier(g)` returns `1/(1+t^2)/pi`
	`f = 2exp(-abs(w)) - 1` `simple(ifourier(f,t))` returns `-(-2+piDirac(t))/(1+t^2)/pi`
	`syms w real` `f = exp(-w^2abs(v))sin(v)/v` `ifourier(f,v,t)` returns `-1/2*(-atan((t+1)/w^2) +atan((-1+t)/w^2))/pi`

See Also

fourier, ilaplace, iztrans

hypergeom ilaplace