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### 28.6 Functions and Variables for Poisson series

Function: intopois (a)

  Converts a into a Poisson encoding.

  Categories:Poisson series ·

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.

  Categories:Poisson series ·

Function: poisdiff (a, b)

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

  Categories:Poisson series ·

Function: poisexpt (a, b)

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

  Categories:Poisson series ·

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.

  Categories:Poisson series ·

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)].

  Categories:Poisson series ·

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.

  Categories:Poisson series ·

Function: poisplus (a, b)

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

  Categories:Poisson series ·

Function: poissimp (a)

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

  Categories:Poisson series ·

Special symbol: poisson

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

  Categories:Poisson series ·

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)`.

  Categories:Poisson series ·

Function: poistimes (a, b)

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

  Categories:Poisson series ·

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.

  Categories:Poisson series ·

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.

  Categories:Poisson series ·Display functions ·

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