## 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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