15.4 Poisson Reihen

Function: intopois (a) ¶

Converts a into a Poisson encoding.

Function: 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.

Function: poisdiff (a, b) ¶

Differentiates a with respect to b. b must occur only in the trig arguments or only in the coefficients.

Function: poisexpt (a, b) ¶

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

Function: 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.

Option variable: 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)].

Function: 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.

Function: poisplus (a, b) ¶

Is functionally identical to intopois (a + b).

Function: poissimp (a) ¶

Converts a into a Poisson series for a in general representation.

Special symbol: poisson ¶

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

Function: 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).

Function: poistimes (a, b) ¶

Is functionally identical to intopois (a*b).

Function: 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.

Function: 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.