Poisson Reihen (Maxima Manual)

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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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