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Calling Sequence
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laplace(expr, t, s)
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Parameters
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expr
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expression, equation, or set of expressions and/or equations to be transformed
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t
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variable expr is transformed with respect to t
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s
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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 laplace function computes the Laplace transform (F(s)) of expr (f(t)) with respect to t, using the definition:
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Expressions involving a wide variety of functions including exponentials, trigonometrics, Bessel functions, error functions, and many others can be transformed.
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The laplace function also recognizes derivatives (diff or Diff) and integrals (int or Int).
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When transforming expressions like diff(y(t), t, s), laplace will insert the initial values , , etc. is the value of the first derivative at 0; is the second derivative at 0, and so on.
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Both laplace and invlaplace recognize the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).
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Users can add their own functions to laplace's internal lookup table by using the addtable function.
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If the option opt is set to 'NO_INT', then the program will not resort to integration of the original problem if all other methods fail. This will increase the speed at which the transform will run.
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The command with(inttrans,laplace) allows the use of the abbreviated form of this command.
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Examples
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laplace(t^2+sin(t)=y(t), t, s);
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laplace(t^(3/2)-exp(t)+sinh(a*t), t, s);
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laplace(diff(y(t), t$2)-y(t)=sin(a*t), t, s-2);
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laplace(BesselI(0,a*t), t, s);
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laplace(Heaviside(t-c)*f(t),t,s);
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laplace(Heaviside(t-c)*f(t),t,s);
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addtable(laplace,myfunc(t),Myfunc(s),t,s):
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laplace(t^3*exp(a*t)*myfunc(4*t),t,w);
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addtable(laplace,myfunc2(t*a)^n,1/((abs(n)+1)/2)!*Myfunc2(s)+a,t,s,{a,n},
n::odd):
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laplace(myfunc2(4*t)^7,t,w);
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Compatibility
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The inttrans[laplace] command was updated in Maple 2019.
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