transform::fourier,
transform::invfourier -- Fourier and inverse Fourier
transform
Introductiontransform::fourier(f, t, s) computes the Fourier
transform int(f*exp(I*s*t), t=-infinity..infinity) of the
expression f = f(t) with respect to the variable
t at the point s.
transform::invfourier(F, S, T) computes the inverse
Fourier transform 1/2/PI* int(F*exp(-I*S*T),
S=-infinity..infinity) of the expression F = F(S) with
respect to the variable S at the point T.
Call(s)transform::fourier(f, t, s)
transform::invfourier(F, S, T)
Parametersf, F |
- | arithmetical expressions |
t, S |
- | identifiers (the transformation variables) |
s, T |
- | arithmetical expressions (the evaluation points) |
Returnsan arithmetical expression
f, F
Related
Functions
Detailstransform::invfourier(F, S, T) is computed as
transform::fourier(F, S, -T)/2/PI.
This result is returned, if no explicit representation of the
transformation is found.numeric::fft and numeric::invfft.
Example
1The following call produces the Fourier transform as an
expression in the variable s:
>> transform::fourier(exp(-t^2), t, s)
/ 2 \
1/2 | s |
PI exp| - -- |
\ 4 /
>> transform::invfourier(%, s, t)
2
exp(- t )
Note that the Fourier transform can be evaluated directly at a specific point such as s = 2*a or s = 5:
>> transform::fourier(t*exp(-a*t^2), t, s), transform::fourier(t*exp(-a*t^2), t, 2*a), transform::fourier(t*exp(-a*t^2), t, 2)
/ 2 \
1/2 | s | 1/2 / 1 \
1/2 I s PI exp| - --- | 1/2 I PI exp| - - |
\ 4 a / I PI exp(-a) \ a /
--------------------------, ---------------, ------------------
3/2 1/2 3/2
a a a
Example
2An unevaluated call is returned, if no explicit representation of the transform is found:
>> transform::fourier(besselJ(0, 1/(1 + t^2)), t, s)
/ / 1 \ \
transform::fourier| besselJ| 0, ------ |, t, s |
| | 2 | |
\ \ t + 1 / /
>> transform::invfourier(%, s, t)
/ 1 \
besselJ| 0, ------ |
| 2 |
\ t + 1 /
Note that the inverse transform is related to the direct transform:
>> transform::invfourier(unknown(s), s, t)
transform::fourier(unknown(s), s, -t)
-------------------------------------
2 PI
Example
3The distribution dirac is handled:
>> transform::fourier(t^3, t, s)
2 I PI dirac(s, 3)
>> transform::invfourier(%, s, t)
3
t
>> transform::fourier(heaviside(t - t0), t, s)
/ I \
exp(I s t0) | PI dirac(s) + - |
\ s /
Example
4The Fourier transform of a function is related to the Fourier transform of its derivative:
>> transform::fourier(diff(f(t), t), t, s)
-I s transform::fourier(f(t), t, s)
Backgroundtransform::invfourier used to be called
transform::ifourier.