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""" Symbolic Series
Symbolic series are special kinds of symbolic expressions that are constructed via the :meth:`Expression.series <sage.symbolic.expression.Expression.series>` method. They usually have an ``Order()`` term unless the series representation is exact, see :meth:`~sage.symbolic.series.SymbolicSeries.is_terminating_series`.
For series over general rings see :class:`power series <sage.rings.power_series_poly.PowerSeries_poly>` and :class:`Laurent series<sage.rings.laurent_series_ring_element.LaurentSeries>`.
EXAMPLES:
We expand a polynomial in `x` about `0`, about `1`, and also truncate it back to a polynomial::
sage: var('x,y') (x, y) sage: f = (x^3 - sin(y)*x^2 - 5*x + 3); f x^3 - x^2*sin(y) - 5*x + 3 sage: g = f.series(x, 4); g 3 + (-5)*x + (-sin(y))*x^2 + 1*x^3 + Order(x^4) sage: g.truncate() x^3 - x^2*sin(y) - 5*x + 3 sage: g = f.series(x==1, oo); g (-sin(y) - 1) + (-2*sin(y) - 2)*(x - 1) + (-sin(y) + 3)*(x - 1)^2 + 1*(x - 1)^3 sage: h = g.truncate(); h (x - 1)^3 - (x - 1)^2*(sin(y) - 3) - 2*(x - 1)*(sin(y) + 1) - sin(y) - 1 sage: h.expand() x^3 - x^2*sin(y) - 5*x + 3
We compute another series expansion of an analytic function::
sage: f = sin(x)/x^2 sage: f.series(x,7) 1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7) sage: f.series(x==1,3) (sin(1)) + (cos(1) - 2*sin(1))*(x - 1) + (-2*cos(1) + 5/2*sin(1))*(x - 1)^2 + Order((x - 1)^3) sage: f.series(x==1,3).truncate().expand() -2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1)
Following the GiNaC tutorial, we use John Machin's amazing formula `\pi = 16 \mathrm{tan}^{-1}(1/5) - 4 \mathrm{tan}^{-1}(1/239)` to compute digits of `\pi`. We expand the arc tangent around 0 and insert the fractions 1/5 and 1/239.
::
sage: x = var('x') sage: f = atan(x).series(x, 10); f 1*x + (-1/3)*x^3 + 1/5*x^5 + (-1/7)*x^7 + 1/9*x^9 + Order(x^10) sage: (16*f.subs(x==1/5) - 4*f.subs(x==1/239)).n() 3.14159268240440
Note: The result of an operation or function of series is not automatically expanded to a series. This must be explicitly done by the user::
sage: ex1 = sin(x).series(x, 4); ex1 1*x + (-1/6)*x^3 + Order(x^4) sage: ex2 = cos(x).series(x, 4); ex2 1 + (-1/2)*x^2 + Order(x^4) sage: ex1 + ex2 (1 + (-1/2)*x^2 + Order(x^4)) + (1*x + (-1/6)*x^3 + Order(x^4)) sage: (ex1 + ex2).series(x,4) 1 + 1*x + (-1/2)*x^2 + (-1/6)*x^3 + Order(x^4) sage: x*ex1 x*(1*x + (-1/6)*x^3 + Order(x^4)) sage: (x*ex1).series(x,5) 1*x^2 + (-1/6)*x^4 + Order(x^5) sage: sin(ex1) sin(1*x + (-1/6)*x^3 + Order(x^4)) sage: sin(ex1).series(x,9) 1*x + (-1/3)*x^3 + 11/120*x^5 + (-53/2520)*x^7 + Order(x^9) sage: (sin(x^2)^(-5)).series(x,3) 1*x^(-10) + 5/6*x^(-6) + 3/8*x^(-2) + 367/3024*x^2 + Order(x^3) sage: (cot(x)^(-3)).series(x,3) Order(x^3) sage: (cot(x)^(-3)).series(x,4) 1*x^3 + Order(x^4)
TESTS:
Check that :trac:`20088` is fixed::
sage: ((1+x).series(x)^pi).series(x,3) 1 + pi*x + (-1/2*pi + 1/2*pi^2)*x^2 + Order(x^3)
Check that :trac:`14878` is fixed, this should take only microseconds::
sage: sin(x*sin(x*sin(x*sin(x)))).series(x,8) 1*x^4 + (-1/6)*x^6 + Order(x^8) sage: sin(x*sin(x*sin(x*sin(x)))).series(x,12) 1*x^4 + (-1/6)*x^6 + (-19/120)*x^8 + (-421/5040)*x^10 + Order(x^12)
Check that :trac:`22959` is fixed::
sage: (x/(1-x^2)).series(x==0, 10) 1*x + 1*x^3 + 1*x^5 + 1*x^7 + 1*x^9 + Order(x^10) sage: (x/(1-x^2)).series(x==0, 11) 1*x + 1*x^3 + 1*x^5 + 1*x^7 + 1*x^9 + Order(x^11) sage: (x^2/(1-x^2)).series(x==0, 10) 1*x^2 + 1*x^4 + 1*x^6 + 1*x^8 + Order(x^10) sage: (x^2/(1-x^2)).series(x==0, 11) 1*x^2 + 1*x^4 + 1*x^6 + 1*x^8 + 1*x^10 + Order(x^11)
Check that :trac:`22733` is fixed::
sage: _ = var('z') sage: z.series(x) (z) """
#***************************************************************************** # Copyright (C) 2015 Ralf Stephan <ralf@ark.in-berlin.de> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
from sage.libs.pynac.pynac cimport * from sage.symbolic.expression cimport Expression, new_Expression_from_GEx
cdef class SymbolicSeries(Expression): def __init__(self, SR): """ Trivial constructor.
EXAMPLES::
sage: loads(dumps((x+x^3).series(x,2))) 1*x + Order(x^2) """ Expression.__init__(self, SR, 0) self._parent = SR
def is_series(self): """ TESTS::
sage: ex = sin(x).series(x,5) sage: ex.is_series() doctest:...: DeprecationWarning: ex.is_series() is deprecated. Use isinstance(ex, sage.symbolic.series.SymbolicSeries) instead See http://trac.sagemath.org/17659 for details. True """
def is_terminating_series(self): """ Return True if the series is without order term.
A series is terminating if it can be represented exactly, without requiring an order term. You can explicitly request terminating series by setting the order to positive infinity.
OUTPUT:
Boolean. ``True`` if the series has no order term.
EXAMPLES::
sage: (x^5+x^2+1).series(x, +oo) 1 + 1*x^2 + 1*x^5 sage: (x^5+x^2+1).series(x,+oo).is_terminating_series() True sage: SR(5).is_terminating_series() False sage: exp(x).series(x,10).is_terminating_series() False """
def truncate(self): """ Given a power series or expression, return the corresponding expression without the big oh.
OUTPUT:
A symbolic expression.
EXAMPLES::
sage: f = sin(x)/x^2 sage: f.truncate() sin(x)/x^2 sage: f.series(x,7) 1*x^(-1) + (-1/6)*x + 1/120*x^3 + (-1/5040)*x^5 + Order(x^7) sage: f.series(x,7).truncate() -1/5040*x^5 + 1/120*x^3 - 1/6*x + 1/x sage: f.series(x==1,3).truncate().expand() -2*x^2*cos(1) + 5/2*x^2*sin(1) + 5*x*cos(1) - 7*x*sin(1) - 3*cos(1) + 11/2*sin(1) """
def default_variable(self): """ Return the expansion variable of this symbolic series.
EXAMPLES::
sage: s=(1/(1-x)).series(x,3); s 1 + 1*x + 1*x^2 + Order(x^3) sage: s.default_variable() x """
def coefficients(self, x=None, sparse=True): r""" Return the coefficients of this symbolic series as a list of pairs.
INPUT:
- ``x`` -- optional variable.
- ``sparse`` -- Boolean. If ``False`` return a list with as much entries as the order of the series.
OUTPUT:
Depending on the value of ``sparse``,
- A list of pairs ``(expr, n)``, where ``expr`` is a symbolic expression and ``n`` is a power (``sparse=True``, default)
- A list of expressions where the ``n``-th element is the coefficient of ``x^n`` when self is seen as polynomial in ``x`` (``sparse=False``).
EXAMPLES::
sage: s=(1/(1-x)).series(x,6); s 1 + 1*x + 1*x^2 + 1*x^3 + 1*x^4 + 1*x^5 + Order(x^6) sage: s.coefficients() [[1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5]] sage: s.coefficients(x, sparse=False) [1, 1, 1, 1, 1, 1] sage: x,y = var("x,y") sage: s=(1/(1-y*x-x)).series(x,3); s 1 + (y + 1)*x + ((y + 1)^2)*x^2 + Order(x^3) sage: s.coefficients(x, sparse=False) [1, y + 1, (y + 1)^2]
""" else: raise ValueError("Cannot return dense coefficient list with noninteger exponents.") raise ValueError("Cannot return dense coefficient list with negative valuation.")
def power_series(self, base_ring): """ Return algebraic power series associated to this symbolic series. The coefficients must be coercible to the base ring.
EXAMPLES::
sage: ex=(gamma(1-x)).series(x,3); ex 1 + euler_gamma*x + (1/2*euler_gamma^2 + 1/12*pi^2)*x^2 + Order(x^3) sage: g=ex.power_series(SR); g 1 + euler_gamma*x + (1/2*euler_gamma^2 + 1/12*pi^2)*x^2 + O(x^3) sage: g.parent() Power Series Ring in x over Symbolic Ring """
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