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Calling Sequence
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fourier(expr, t, w)
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Parameters
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expr
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expression, equation, or set of equations and/or expressions to be transformed
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t
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variable expr is transformed with respect to t
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w
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parameter of transform
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opt
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option to run this under (optional)
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Description
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The fourier function computes the Fourier transform (F(w)) of expr (f(t)) with respect to t, using the definition
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Expressions involving complex exponentials, polynomials, trigonometrics (sin, cos) and a variety of functions and other integral transforms can be transformed.
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The fourier function recognizes derivatives (diff or Diff) and integrals (int or Int).
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Users can add their own functions to fourier's internal lookup table with the function inttrans[addtable].
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fourier recognizes the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).
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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.
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The command with(inttrans,fourier) allows the use of the abbreviated form of this command.
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Examples
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fourier(3/(a^2 + t^2),t,w);
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| (1) |
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fourier(diff(f(x),x$4),x,w);
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F:= int(g(x)*h(t-x),x=-infinity..infinity):
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fourier(t*exp(-3*t)*Heaviside(t),t,w);
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fourier(1/(4 - I*t)^(1/3),t,2+w);
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| (5) |
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fourier(diff(y(t), t$2)-y(t)=sin(a*t), t, s);
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fourier(BesselJ(0,4*(t^2 + 1)^(1/2)), t, s);
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| (7) |
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addtable(fourier,myfunc(t),Myfunc(s)/(1+s^2),t,s):
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fourier(exp(3*I*t)*myfunc(2*t),t,w);
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| (8) |
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Compatibility
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The inttrans[fourier] command was updated in Maple 2019.
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