Maxima – 168 – Somme, prodotti e serie – 5

dtsf

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 forb 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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