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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.


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