Documentation |
Generalized series expansions
This functionality does not run in MATLAB.
Series::gseries(f, x, <order>, <Left | Right>) Series::gseries(f, x = a, <order>, <Left | Right>)
Series::gseries is the domain of series expansions generalizing Taylor, Laurent and Puiseux expansions.
The call Series::gseries(f, x) computes a series expansion at the right hand side of x = 0.
The system functions series and asympt are the main application of this domain. The latter function only returns elements of this domain, whereas series can return an element of Series::gseries in cases, where a Puiseux series expansion does not exist.
There may be no need to explicitly create elements of this domain, but to work with the results of the mentioned system functions.
See the help page of the system function asympt for a detailed description of the parameters and examples for working with elements of the domain Series::gseries.
The function is sensitive to the global variable ORDER, which determines the default number of terms of the expansion.
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 |
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). |
Calling an element of Series::gseries as a function yields the object itself, regardless of the arguments. The arguments are not evaluated.
Series::gseries implements standard arithmetic of generalized series expansions. Use the ordinary arithmetical operators +, -, *, /, and ^.
The system functions coeff, lcoeff, nthcoeff, lterm, nthterm, lmonomial, nthmonomial, and ldegree work on generalized series expansions. See the corresponding help pages of these functions for calling parameters. See the description of these methods below for further details.
The method "indet" returns the series variable of the series expansion, i.e., if s is an object of the domain Series::gseries, then s::dom::indet(s) returns the series variable.
The method "point" returns the expansion point of the series.
Use the function expr to convert a generalized series expansion into an arithmetical expression (as an element of a kernel domain).
A series of the domain type Series::gseries consists of four operands:
A list of pairs [c_{i}, f_{i}]. Each pair represents a monomialc_{i} f_{i} of the series expansion, where the c_{i} are the coefficients and f_{i} the terms of s. The coefficients do not contain the series variable.
This list can be empty, if the order of the expansion is zero.
An arithmetical expression g representing the error term of the form O(g). It may be the integer 0, in which case the expansion is exact.
The series variable x.
The expansion point a.