## Maxima – 168 – Somme, prodotti e serie – 5

Continuo da qui, copio dal Reference Manual, PDF scaricabile da qui, sono a p.509.

Serie di Fourier

The `fourie` package comprises functions for the symbolic computation of Fourier series. There are functions in the `fourie` package to calculate Fourier integral coefficients and some functions for manipulation of expressions.

Nota: senza la “`r`“.

`equalp (x, y)`
Returns `true` if `equal (x, y)` otherwise `false` (doesn’t give an error message like `equal (x, y)` would do in this case).

`remfun (f, expr)`
`remfun (f, expr, x)`
`remfun (f, expr)` replaces all occurrences of `f (arg)` by `arg in expr.`

`remfun (f, expr, x)` replaces all occurrences of `f (arg)` by `arg` in `expr` only if `arg` contains the variable `x`.

`funp (f, expr)`
`funp (f, expr, x)`
`funp (f, expr)` returns `true` if `expr` contains the function `f`.

`funp (f, expr, x)` returns `true` if `expr` contains the function `f` and the variable `x` is somewhere in the argument of one of the instances of `f`.

`absint (f, x, halfplane)`
`absint (f, x)`
`absint (f, x, a, b)`
`absint (f, x, halfplane)` returns the indefinite integral of `f` with respect to `x` in the given `halfplane` (`pos`, `neg`, or `both`). `f` may contain expressions of the form `abs (x)`, `abs (sin (x))`, `abs (a) * exp (-abs (b) * abs (x))`.

`absint (f, x)` is equivalent to `absint (f, x, pos)`.

`absint (f, x, a, b)` returns the definite integral of `f` with respect to `x` from `a` to `b`. `f` may include absolute values.

`fourier (f, x, p)`
Returns a list of the Fourier coefficients of `f(x)` defined on the interval `[-p, p]`.

`foursimp (l)`
Simplifies `sin (n %pi)` to `0` if `sinnpiflag` is `true` and `cos (n %pi)` to `(-1)^n` if `cosnpiflag` is `true`.

`sinnpiflag`
Default value: `true`. See `foursimp`.

`cosnpiflag`
Default value: `true`. See `foursimp`.

`fourexpand (l, x, p, limit)`
Constructs and returns the Fourier series from the list of Fourier coefficients `l` up through `limit` terms (`limit` may be `inf`). `x` and `p` have same meaning as in `fourier`.

`fourcos (f, x, p)`
Returns the Fourier cosine coefficients for `f(x)` defined on `[0, p]`.

`foursin (f, x, p)`
Returns the Fourier sine coefficients for `f(x)` defined on `[0, p]`.

`totalfourier (f, x, p)`
Returns `fourexpand (foursimp (fourier (f, x, p)), x, p, 'inf)`.

`fourint (f, x)`
Constructs and returns a list of the Fourier integral coefficients of `f(x)` defined on `[minf, inf]`.

`fourintcos (f, x)`
Returns the Fourier cosine integral coefficients for `f(x)` on `[0, inf]`.

`fourintsin (f, x)`
Returns the Fourier sine integral coefficients for `f(x)` on `[0, inf]`.

Funzioni e variabili per la serie di Poisson

`intopois (a)`
Converts `a` into a Poisson encoding.

`outofpois (a)`
Converts `a` from Poisson encoding to general representation. If `a` is not in Poisson form, `outofpois` carries out the conversion, i.e., the return value is `outofpois (intopois (a))`. This function is thus a canonical simplifier for sums of powers of sine and cosine terms of a particular type.

`poisdiff (a, b)`
Differentiates `a` with respect to `b`. `b` must occur only in the trig arguments or only in the coefficients.

`poisexpt (a, b)`
Functionally identical to `intopois (a^b)`. `b` must be a positive integer.

`poisint (a, b)`
Integrates in `a` similarly restricted sense (to `poisdiff`). Non-periodic terms in `b` are dropped if `b` is in the trig arguments.

`poislim`
Default value: `5`.

`poislim` determines the domain of the coefficients in the arguments of the trig functions. The initial value of `5` corresponds to the interval `[-2^(5-1)+1,2^(5-1)]`, or `[-15,16]`, but it can be set to `[-2^(n-1)+1, 2^(n-1)]`.

`poismap (series, sinfn, cosfn)`
will map the functions `sinfn` on the sine terms and `cosfn` on the cosine terms of the Poisson `series` given. `sinfn` and `cosfn` are functions of two arguments which are a coefficient and a trigonometric part of a term in series respectively.

`poisplus (a, b)`
Is functionally identical to `intopois (a + b)`.

`poissimp (a)`
Converts `a` into a Poisson series for a in general representation.

`poisson`
The symbol `/P/` follows the line label of Poisson series expressions.

`poissubst (a, b, c)`
Substitutes `a` for`b` in `c`. `c` is a Poisson series.

(1) Where `b` is a variable `u`, `v`, `w`, `x`, `y`, or `z`, then `a` must be an expression linear in those variables (e.g., `6*u + 4*v`).

(2) Where `b` is other than those variables, then `a` must also be free of those variables, and furthermore, free of sines or cosines.

`poissubst (a, b, c, d, n)` is a special type of substitution which operates on `a` and `b` as in type (1) above, but where `d` is a Poisson series, expands `cos(d)` and `sin(d)` to order `n` so as to provide the result of substituting `a + d` for `b` in `c`. The idea is that `d` is an expansion in terms of a small parameter. For example, `poissubst (u, v, cos(v), %e, 3)` yields `cos(u)*(1 - %e^2/2) - sin(u)*(%e - %e^3/6)`.

`poistimes (a, b)`
Is functionally identical to `intopois (a*b)`.

`poistrim ()`
is a reserved function name which (if the user has defined it) gets applied during Poisson multiplication. It is a predicate function of 6 arguments which are the coefficients of the `u, v, ..., z` in `a` term. Terms for which `poistrim` is `true` (for the coefficients of that term) are eliminated during multiplication.

`printpois (a)`
Prints a Poisson series in a readable format. In common with `outofpois`, it will convert `a` into a Poisson encoding first, if necessary.

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