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Series::gseries

Generalized series expansions

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Syntax

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

Description

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.

    Note:   Note that elements of Series::gseries only represents directional (real) series expansions.

Environment Interactions

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

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).

Return Values

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

Function Calls

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

Operations

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).

Operands

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

  1. A list of pairs [ci, fi]. Each pair represents a monomialcifi of the series expansion, where the ci are the coefficients and fi 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.

  2. 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.

  3. The series variable x.

  4. The expansion point a.

Methods

expand all

Mathematical Methods

_divide(s, t)

If the arguments are not of domain type Series::gseries, then they are converted into such objects. FAIL is returned, if one of these conversions fails.

This method overloads the function _divide for elements of Series::gseries, i.e., you may use it in the form s/t.

_invert(s)

This method overloads the function _invert for elements of Series::gseries, i.e., you may use it in the form 1/s.

_mult(s, t, …)

If both s and t are series expansions of the domain Series::gseries, then the result is a series expansion of the domain Series::gseries, too. Both series expansions must have the same series variable and expansion point, otherwise FAIL is returned.

If s or t is a series expansion of the domain Series::Puiseux, then it is converted into an object of Series::gseries. If this fails, then FAIL is returned. Otherwise, the product is computed and returned as an object of the domain Series::gseries.

If s is a series expansion and t is an arithmetical expression, then t is converted into a series expansion via the constructor Series::gseries (and vice versa).

Each argument of this method that is not of the domain type Series::gseries is converted into such an element, i.e., a generalized series expansion is computed. If this fails, then FAIL is returned.

This method overloads the function _mult for elements of Series::gseries, i.e., you may use it in the form s*t*....

_negate(s)

This method overloads the function _negate for elements of Series::gseries, i.e., you may use it in the form -s.

_plus(s, t, …)

If both s and t are series expansions of the domain Series::gseries, then the result is a series expansion of the domain Series::gseries, too. Both series expansions must have the same series variable and expansion point, otherwise FAIL is returned.

If s or t is a series expansion of the domain Series::Puiseux, then it is converted into an object of Series::gseries. If this fails, then FAIL is returned. Otherwise, the sum is computed and returned as an object of the domain Series::gseries.

If s is a series expansion and t is an arithmetical expression, then t is converted into a series expansion via the constructor Series::gseries (and vice versa).

Each argument of this method that is not of the domain type Series::gseries is converted into such an element, i.e., a generalized series expansion is computed. If this fails, then FAIL is returned.

This method overloads the function _plus for elements of Series::gseries, i.e., you may use it in the form s+t+ ....

_power(s, n)

The exponent n must not involve the series variable of s. Otherwise, an error occurs.

If n is a positive integer, then repeated squaring is used for computing the nth power of s. Otherwise, the binomial theorem is applied after factoring out the leading monomial.

This method overloads the function _power for elements of Series::gseries, i.e., you may use it in the form s^n.

_subtract(s, t)

If the arguments are not of domain type Series::gseries, then they are converted into such objects. FAIL is returned, if one of these conversions fails.

This method overloads the function _subtract for elements of Series::gseries, i.e., you may use it in the form s-t.

Access Methods

coeff(s, <n>)

This method overloads the function coeff for elements of Series::gseries.

Series::gseries::indet(s)

Use the method "point" to get the expansion point of s.

iszero(s)

This method overloads the function iszero for elements of Series::gseries.

lcoeff(s)

This method overloads the function lcoeff for elements of Series::gseries.

ldegree(s)

This method overloads the function ldegree for elements of Series::gseries.

lmonomial(s)

This method overloads the function lmonomial for elements of Series::gseries.

lterm(s)

This method overloads the function lterm for elements of Series::gseries.

nthcoeff(s, n)

This method overloads the function nthcoeff for elements of Series::gseries.

nthmonomial(s, n)

This method overloads the function nthmonomial for elements of Series::gseries.

nthterm(s, n)

This method overloads the function nthterm for elements of Series::gseries.

Series::gseries::point(s)

Use the method "indet" to get the series variable of s.

Conversion Methods

Series::gseries::convert(x)

Series::gseries::convert_to(s, T)

T might be the domain DOM_POLY, where the sum of monomials is considered as a polynomial in the indeterminates of the third operand of s.

If T is the domain DOM_EXPR, then the conversion is the same as implemented by the method "expr" (see below).

If T is the domain Series::Puiseux, then the system tries to convert s into a Puiseux series. If the conversion is not possible, FAIL is returned.

Use the function expr to convert s into an object of a kernel domain.

Series::gseries::create(list, errorTerm, x = a)

    Note:   This method should be used with caution, because no argument checking is performed. Use it to create, not to compute elements of Series::gseries.

expr(s)

This method overloads the function expr for elements of Series::gseries.

series(s, x | x = x0, <order>, <dir>)

This method overloads the function series for elements of Series::gseries. See the corresponding help page for a description of the possible arguments.

Technical Methods

combine(s, <target>)

This method overloads the system function combine. See the corresponding help page for a description of the optional argument target.

has(s, t)

This method overloads the system function has.

map(s, func, …)

This method overloads the function map for elements of Series::gseries.

print(s)

subs(s, x = a, …)

This method overloads the function subs for elements of Series::gseries.

Series::gseries::TeX(s)

This method is called by the system function generate::TeX.

See Also

MuPAD Domains

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