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Series::gseries -- the domain of generalized series expansions

Introduction

Series::gseries is the domain of series expansions generalizing Taylor, Laurent and Puiseux expansions.

Creating Elements

Series::gseries(f, x <, order>)
Series::gseries(f, x = a <, order>)
Series::gseries(f, x = a <, order> <, Right>)
Series::gseries(f, x = a <, order> <, Left>)

Parameters

f - an arithmetical expression
x - the series variable: an identifier
a - the expansion point: an arithmetical expression or +/-infinity
order - the truncation order: a nonnegative integer

Options

Left - compute a series expansion that is valid for real x smaller than a.
Right - compute a series expansion that is valid for real x larger than a (the default case).

Returns

an object of domain type Series::gseries, or the value FAIL.

Side Effects

The function is sensitive to the global variable ORDER, which determines the default number of terms of the expansion.

Details

Operands

A series of the domain type Series::gseries consists of three operands:

  1. A list of sublists [ci, fi] of length 2. Each sublist represents a monomial ci*fi of the series expansion, where the ci are the coefficients and fi the terms of s.
  2. The order term of the form O(g(x)), possibly the integer 0, if the expansion is exact.
  3. An arithmetical expression e(x) depending of the series variable x such that e(x) -> infinity for x -> a+.


Related Domains

Series::Puiseux

Important Operations

Series::gseries implements addition and multiplication of generalized series expansions. Use the ordinary arithmetical operators + and *.

The system functions coeff, nthterm, lterm, nthmonomial and lmonomial as well as ldegree work on generalized series expansions.

Use the function expr to convert a generalized series expansion into an arithmetical expression (as an element of a kernel domain).

Result of Evaluation

Evaluating an object of type Series::gseries returns itself.

Function Call

Calling an element of Series::gseries as a function yields the object itself, regardless of the arguments. The arguments are not evaluated.

Method _mult: multiply series expansions

Method _plus: add series expansions

Method _power: the integer power of a series expansions

Method coeff: extract coefficients

Method iszero: test on zero

Method lcoeff: the leading coefficient

Method ldegree: the leading degree

Method lmonomial: the leading monomial

Method lterm: the leading term

Method nthmonomial: extract monomials

Method nthterm: extract terms

Method map: map a function to coefficients

Method subs: substitute subexpressions of monomials

Method subsex: substitute subexpressions of monomials (extended)

Method convert: convert an object into a generalized series expansion

Method convert_to: convert a generalized series expansion into other domains

Method create: create simple and fast a generalized series expansion

Method expr: convert a generalized series expansion into an element of a kernel domain

Method print: pretty-print routine

Method TeX: LaTeX formatting




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