Series
::gseries
Generalized series expansions
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.
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:
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
.
Methods
expand all
Mathematical Methods
_divide
— Divide two series expansions
_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
— Multiplicative inverse of a series expansion
_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
— Multiply series expansions
_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
— Negative of a series expansion
_negate(s
)
This method overloads the function _negate
for elements of Series::gseries
,
i.e., you may use it in the form s
.
_plus
— Add series expansions
_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
— Exponentiation of a series expansion
_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 n
th
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
— Subtract two series expansions
_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 st
.
Access Methods
coeff
— Extract coefficients
coeff(s
, <n
>)
This method overloads the function coeff
for elements of Series::gseries
.
indet
— Serie variable
Series::gseries::indet(s
)
Use the method "point"
to get the expansion
point of s
.
iszero
— Zero test
iszero(s
)
This method overloads the function iszero
for elements of Series::gseries
.
lcoeff
— Leading coefficient
lcoeff(s
)
This method overloads the function lcoeff
for elements of Series::gseries
.
ldegree
— Leading degree
ldegree(s
)
This method overloads the function ldegree
for elements of Series::gseries
.
lmonomial
— Leading monomial
lmonomial(s
)
This method overloads the function lmonomial
for elements of Series::gseries
.
lterm
— Leading term
lterm(s
)
This method overloads the function lterm
for elements of Series::gseries
.
nthcoeff
— Extract a coefficient
nthcoeff(s
, n
)
This method overloads the function nthcoeff
for elements of Series::gseries
.
nthmonomial
— Extract a monomial
nthmonomial(s
, n
)
This method overloads the function nthmonomial
for elements of Series::gseries
.
nthterm
— Extract a term
nthterm(s
, n
)
This method overloads the function nthterm
for elements of Series::gseries
.
point
— Expansion point
Series::gseries::point(s
)
Use the method "indet"
to get the series
variable of s
.
Conversion Methods
convert
— Convert an object into a generalized series expansion
Series::gseries::convert(x
)
convert_to
— Convert a generalized series expansion into other domains
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.
create
— Create simple and fast a generalized series expansion
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
— Convert a generalized series expansion into an element of a kernel domain
expr(s
)
This method overloads the function expr
for elements of Series::gseries
.
series
— Apply the function series
to a generalized series expansion
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
— Apply the function combine
to all terms
combine(s
, <target
>)
This method overloads the system function combine
. See the corresponding help page
for a description of the optional argument target
.
has
— Check whether an object occurs syntactically
has(s
, t
)
This method overloads the system function has
.
map
— Map a function to the coefficients
map(s
, func, …
)
This method overloads the function map
for elements of Series::gseries
.
print
— Prettyprint routine
subs
— Substitute into a generalized series expansion
subs(s
, x = a, …
)
This method overloads the function subs
for elements of Series::gseries
.