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inttrans

  

fourier

  

Fourier transform

 

Calling Sequence

Parameters

Description

Examples

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

withinttrans:

assume0<a&colon;

fourier3a2&plus;t2&comma;t&comma;w

3&pi;&ExponentialE;a~wHeavisidew&plus;&ExponentialE;a~wHeavisidewa~

(1)

fourier&DifferentialD;4&DifferentialD;x4fx&comma;x&comma;w

w4fourierfx&comma;x&comma;w

(2)

F&int;&infin;&infin;gxhtx&DifferentialD;x&colon;

fourier3F&comma;t&comma;w

3fouriergt&comma;t&comma;wfourierht&comma;t&comma;w

(3)

fouriert&ExponentialE;3tHeavisidet&comma;t&comma;w

13&plus;Iw2

(4)

fourier14It13&comma;t&comma;2&plus;w

3&Gamma;23&ExponentialE;84wHeaviside2&plus;w2&plus;w2&sol;3

(5)

fourier&DifferentialD;2&DifferentialD;t2ytyt&equals;sinat&comma;t&comma;s

s2&plus;1fourieryt&comma;t&comma;s&equals;I&pi;Diracs&plus;a~Diracsa~

(6)

fourierBesselJ0&comma;4t2&plus;112&comma;t&comma;s

8&ExponentialE;Iscoss216Heavisides&plus;4Heavisides4s2&plus;16

(7)

addtablefourier&comma;myfunct&comma;Myfuncs1&plus;s2&comma;t&comma;s&colon;

fourier&ExponentialE;3Itmyfunc2t&comma;t&comma;w

2Myfunc12w32w26w&plus;13

(8)

See Also

dsolve

fourier_in_maple

inttrans

inttrans[addtable]

inttrans[invfourier]

 


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