/usr/share/axiom-20170501/src/algebra/ULSCAT.spad is in axiom-source 20170501-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | )abbrev category ULSCAT UnivariateLaurentSeriesCategory
++ Author: Clifton J. Williamson
++ Date Created: 21 December 1989
++ Date Last Updated: 20 September 1993
++ Description:
++ \spadtype{UnivariateLaurentSeriesCategory} is the category of
++ Laurent series in one variable.
UnivariateLaurentSeriesCategory(Coef) : Category == SIG where
Coef : Ring
I ==> Integer
NNI ==> NonNegativeInteger
Term ==> Record(k:I,c:Coef)
SIG ==> UnivariatePowerSeriesCategory(Coef,Integer) with
series : Stream Term -> %
++ \spad{series(st)} creates a series from a stream of non-zero terms,
++ where a term is an exponent-coefficient pair. The terms in the
++ stream should be ordered by increasing order of exponents.
multiplyCoefficients : (I -> Coef,%) -> %
++ \spad{multiplyCoefficients(f,sum(n = n0..infinity,a[n] * x**n)) =
++ sum(n = 0..infinity,f(n) * a[n] * x**n)}.
++ This function is used when Puiseux series are represented by
++ a Laurent series and an exponent.
if Coef has IntegralDomain then
rationalFunction : (%,I) -> Fraction Polynomial Coef
++ \spad{rationalFunction(f,k)} returns a rational function
++ consisting of the sum of all terms of f of degree <= k.
++
++X w:SparseUnivariateLaurentSeries(Fraction(Integer),'z,0):=0
++X rationalFunction(w,0)
rationalFunction : (%,I,I) -> Fraction Polynomial Coef
++ \spad{rationalFunction(f,k1,k2)} returns a rational function
++ consisting of the sum of all terms of f of degree d with
++ \spad{k1 <= d <= k2}.
if Coef has Algebra Fraction Integer then
integrate : % -> %
++ \spad{integrate(f(x))} returns an anti-derivative of the power
++ series \spad{f(x)} with constant coefficient 1.
++ We may integrate a series when we can divide coefficients
++ by integers.
if Coef has integrate: (Coef,Symbol) -> Coef and _
Coef has variables: Coef -> List Symbol then
integrate : (%,Symbol) -> %
++ \spad{integrate(f(x),y)} returns an anti-derivative of the power
++ series \spad{f(x)} with respect to the variable \spad{y}.
if Coef has TranscendentalFunctionCategory and _
Coef has PrimitiveFunctionCategory and _
Coef has AlgebraicallyClosedFunctionSpace Integer then
integrate : (%,Symbol) -> %
++ \spad{integrate(f(x),y)} returns an anti-derivative of
++ the power series \spad{f(x)} with respect to the variable
++ \spad{y}.
RadicalCategory
--++ We provide rational powers when we can divide coefficients
--++ by integers.
TranscendentalFunctionCategory
--++ We provide transcendental functions when we can divide
--++ coefficients by integers.
if Coef has Field then Field
--++ Univariate Laurent series over a field form a field.
--++ In fact, K((x)) is the quotient field of K[[x]].
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