algcurves[algfun_series_sol] - determine Puiseux expansions of an algebraic function with "nice" coefficients
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Calling Sequence
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algfun_series_sol(F, x, y, opts)
algfun_series_sol(F, x=a, y, opts)
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Parameters
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F
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irreducible polynomial in two variables
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x
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name of the independent variable
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y
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-
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name of the dependent variable
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opts
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sequence of optional arguments of the form 'option'='value', where option is one of coeffs, msparse_order, mpoints_search, or RNF
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x=a
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point of expansion
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Description
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The command builds a linear ordinary differential equation which the given algebraic function satisfies. Then it uses the package Slode for finding "nice" points and for building "nice" expansions for the constructed linear ordinary differential equation. The "nice" Puiseux expansion at the point x=a is represented in one of the following forms:
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1.
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For a polynomial, rational, hypergeometric, or d'Alembertian sequence :
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2.
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For an m-sparse m-hypergeometric sequence :
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3.
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For an m-sparse sequence :
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where is a homogeneous linear recurrence with polynomial coefficients for the sequence .
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Options
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Each optional argument is of the type 'option' = 'value'. The following options are supported.
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Specifies the type of the coefficients; the default value is hypergeom.
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–
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dAlembertian, that is, is annihilated by a linear recurrence operator that can be written as a composition of first-order operators (see LinearOperators).
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Specifies the sparse order. This option is given as 'msparse_order'=n, where n is an integer greater than 1. This option is useful only if one of the options 'coeffs'='mhypergeom' or 'coeffs'='sparse' is given.
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Specifies the algorithm for finding the "nice" points. This option is useful only in the case when the point of expansion is not specified and one of the options 'coeffs'='mhypergeom' or 'coeffs'='sparse' is given. The default value is full.
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full, that is, the algorithm, which gives all "nice" points
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quick, that is, the algorithm, which may miss some "nice" points but takes considerably less time than the full algorithm
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Specifies the representation of . This option makes sense only if one of the options 'coeffs'='hypergeom' or 'coeffs'='mhypergeom' is given. The default value is false.
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false, that is, is represented as in the package LREtools.
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true, that is, is the first minimal representation of a hypergeometric term (see RationalNormalForms).
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Examples
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>
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See Also
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algcurves, algcurves[puiseux], algcurves[singularities], LinearOperators, LREtools, LREtools[hypergeomsols], RationalNormalForms, RationalNormalForms[MinimalRepresentation], Slode, Slode[candidate_mpoints], Slode[candidate_points]
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