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inttrans

  

fourier

  

Fourier transform

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

fourier(expr, t, w)

Parameters

expr

-

expression, equation, or set of equations and/or expressions to be transformed

t

-

variable expr is transformed with respect to t

w

-

parameter of transform

opt

-

option to run this under (optional)

Description

• 

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

Fw=ftⅇ−Iwtⅆt

• 

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

• 

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

• 

Users can add their own functions to fourier's internal lookup table with the function inttrans[addtable].

• 

fourier  recognizes the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).

• 

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

• 

For information on computing Fourier transforms on signal data, see Fourier Transforms in Maple.

Examples

with(inttrans):

assume(a>0):

fourier(3/(a^2 + t^2),t,w);

3πHeavisidewⅇa~w+ⅇa~wHeavisidewa~

(1)

fourier(diff(f(x),x$4),x,w);

w4fourierfx,x,w

(2)

F:= int(g(x)*h(t-x),x=-infinity..infinity):

fourier(3*F,t,w);

3fouriergt,t,wfourierht,t,w

(3)

fourier(t*exp(-3*t)*Heaviside(t),t,w);

13+Iw2

(4)

fourier(1/(4 - I*t)^(1/3),t,2+w);

3Γ23ⅇ84wHeaviside2+w2+w23

(5)

fourier(diff(y(t), t$2)-y(t)=sin(a*t), t, s);

s2+1fourieryt,t,s=IπDiracs+a~+Diracs+a~

(6)

fourier(BesselJ(0,4*(t^2 + 1)^(1/2)), t, s);

8ⅇIscoss216Heavisides+4Heavisides4s2+16

(7)

addtable(fourier,myfunc(t),Myfunc(s)/(1+s^2),t,s):

fourier(exp(3*I*t)*myfunc(2*t),t,w);

2Myfuncw232w26w+13

(8)

Compatibility

• 

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

See Also

dsolve

fourier_in_maple

inttrans

inttrans[addtable]

inttrans[invfourier]