Maxima – 113 – Differenziazione – 4

Continuo da qui, copio dal Reference Manual, PDF scaricabile da qui, sono a p.325.

gradef (f (x_1, ..., x_n) , g_1, ..., g_m)
gradef (f (a, x, expr)
Defines the partial derivatives (i.e., the components of the gradient) of the function f or variable a.

gradef (f(x_1, ..., x_n), g_1, ..., g_m) defines df/dx_i as g_i, where g_i is an expression; g_i may be a function call, but not the name of a function. The number of partial derivatives m may be less than the number of arguments n, in which case derivatives are defined with respect to x_1 through x_m only.

gradef (a, x, expr) defines the derivative of variable a with respect to x as expr. This also establishes the dependence of a on x (via depends(a, x)).

The first argument f(x_1, ..., x_n) or a is quoted, but the remaining arguments g_1, ..., g_m are evaluated. gradef returns the function or variable for which the partial derivatives are defined.

gradef can redefine the derivatives of Maxima’s built-in functions. For example, gradef(sin(x), sqrt (1 - sin(x)^2)) redefines the derivative of sin.

gradef cannot define partial derivatives for a subscripted function.

printprops ([f_1, ..., f_n], gradef) displays the partial derivatives of the functions f_1, ..., f_n, as defined by gradef.

printprops ([a_n, ..., a_n], atomgrad) displays the partial derivatives of the variables a_n, ..., a_n, as defined by gradef. Forse a_1, ..., a_n.

gradefs is the list of the functions for which partial derivatives have been defined by gradef. gradefs does not include any variables for which partial derivatives have been defined by gradef.

Gradients are needed when, for example, a function is not known explicitly but its first derivatives are and it is desired to obtain higher order derivatives.

gradefs
Default value: [].

gradefs is the list of the functions for which partial derivatives have been defined by gradef. gradefs does not include any variables for which partial derivatives have been defined by gradef.

laplace (expr, t, s)
Attempts to compute the Laplace transform of expr with respect to the variable t and transform parameter s.

laplace recognizes in expr the functions delta, exp, log, sin, cos, sinh, cosh, and erf, as well as derivative, integrate, sum, and ilt. If laplace fails to find a transform the function specint is called. specint can find the laplace transform for expressions with special functions like the bessel functions bessel_j, bessel_i, ... and can handle the unit_step function. See also specint.

If specint cannot find a solution too, a noun laplace is returned.

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 ilt, the inverse Laplace transform.

(%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
(%i5) assume(a>0)$

(%i6) laplace(gamma_incomplete(a,t),t,s),gamma_expand:true;
Is a an integer?

y;
                                             (- a) - 1
                        gamma(a)   gamma(a) s
(%o6)                   -------- - -------------------
                           s             1     a
                                        (- + 1)
                                         s
(%i7) factor(laplace(gamma_incomplete(1/2,t),t,s));
                                              s + 1
                      sqrt(%pi) (sqrt(s) sqrt(-----) - 1)
                                                s
(%o7)                 -----------------------------------
                                3/2      s + 1
                               s    sqrt(-----)
                                           s
(%i8) assume(exp(%pi*s)>1)$

(%i9) laplace(sum((-1)^n*unit_step(t-n*%pi)*sin(t),n,0,inf),t,s), simpsum;
                                      %pi s
                                    %e
(%o9)                      -------------------------
                             2        %pi s    2
                           (s  + 1) %e      - s  - 1
(%i10) factor(%);
                                     %pi s
                                   %e
(%o10)                      ----------------------
                              2         %pi s
                            (s  + 1) (%e      - 1)

%o9 nella reference è diversa, non fattorializzata.

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