Maxima Function
laplace (expr, t, s)
Attempts to compute the Laplace transform of expr with respect to the variable t
and transform parameter s.
If laplace
cannot find a solution, a noun 'laplace
is returned.
laplace
recognizes in expr the functions
delta
, exp
, log
, sin
, cos
, sinh
, cosh
, and erf
,
as well as derivative
, integrate
, sum
, and ilt
.
If some other functions are present,
laplace
may not be able to compute the transform.
expr may also be a linear, constant coefficient differential equation in
which case atvalue
of the dependent variable is used.
The required atvalue may be supplied either before or after the transform is computed. Since the initial conditions must be specified at zero, if one has boundary conditions imposed elsewhere he can impose these on the general solution and eliminate the constants by solving the general solution for them and substituting their values back.
laplace
recognizes convolution integrals of the form
integrate (f(x) * g(t - x), x, 0, t)
;
other kinds of convolutions are not recognized.
Functional relations must be explicitly represented in expr;
implicit relations, established by depends
, are not recognized.
That is, if f depends on x and y,
f (x, y)
must appear in expr.
See also , the inverse Laplace transform.
Examples:
(%i1) laplace (exp (2*t + a) * sin(t) * t, t, s); a %e (2 s - 4) (%o1) --------------- 2 2 (s - 4 s + 5) (%i2) laplace ('diff (f (x), x), x, s); (%o2) s laplace(f(x), x, s) - f(0) (%i3) diff (diff (delta (t), t), t); 2 d (%o3) --- (delta(t)) 2 dt (%i4) laplace (%, t, s); ! d ! 2 (%o4) - -- (delta(t))! + s - delta(0) s dt ! !t = 0