Power Series¶

Sage provides an implementation of dense and sparse power series over any Sage base ring. This is the base class of the implementations of univariate and multivariate power series ring elements in Sage (see also Power Series Methods, Multivariate Power Series).

AUTHORS:

William Stein
David Harvey (2006-09-11): added solve_linear_de() method
Robert Bradshaw (2007-04): sqrt, rmul, lmul, shifting
Robert Bradshaw (2007-04): Cython version
Simon King (2012-08): use category and coercion framework, Issue #13412

EXAMPLES:

Sage

sage: R.<x> = PowerSeriesRing(ZZ)
sage: TestSuite(R).run()
sage: R([1,2,3])
1 + 2*x + 3*x^2
sage: R([1,2,3], 10)
1 + 2*x + 3*x^2 + O(x^10)
sage: f = 1 + 2*x - 3*x^3 + O(x^4); f
1 + 2*x - 3*x^3 + O(x^4)
sage: f^10
1 + 20*x + 180*x^2 + 930*x^3 + O(x^4)
sage: g = 1/f; g
1 - 2*x + 4*x^2 - 5*x^3 + O(x^4)
sage: g * f
1 + O(x^4)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> TestSuite(R).run()
>>> R([Integer(1),Integer(2),Integer(3)])
1 + 2*x + 3*x^2
>>> R([Integer(1),Integer(2),Integer(3)], Integer(10))
1 + 2*x + 3*x^2 + O(x^10)
>>> f = Integer(1) + Integer(2)*x - Integer(3)*x**Integer(3) + O(x**Integer(4)); f
1 + 2*x - 3*x^3 + O(x^4)
>>> f**Integer(10)
1 + 20*x + 180*x^2 + 930*x^3 + O(x^4)
>>> g = Integer(1)/f; g
1 - 2*x + 4*x^2 - 5*x^3 + O(x^4)
>>> g * f
1 + O(x^4)

Sage Live

R.<x> = PowerSeriesRing(ZZ)
TestSuite(R).run()
R([1,2,3])
R([1,2,3], 10)
f = 1 + 2*x - 3*x^3 + O(x^4); f
f^10
g = 1/f; g
g * f

In Python (as opposed to Sage) create the power series ring and its generator as follows:

Sage

sage: R = PowerSeriesRing(ZZ, 'x')
sage: x = R.gen()
sage: parent(x)
Power Series Ring in x over Integer Ring

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, 'x')
>>> x = R.gen()
>>> parent(x)
Power Series Ring in x over Integer Ring

Sage Live

R = PowerSeriesRing(ZZ, 'x')
x = R.gen()
parent(x)

EXAMPLES:

This example illustrates that coercion for power series rings is consistent with coercion for polynomial rings.

Sage

sage: poly_ring1.<gen1> = PolynomialRing(QQ)
sage: poly_ring2.<gen2> = PolynomialRing(QQ)
sage: huge_ring.<x> = PolynomialRing(poly_ring1)

Python

>>> from sage.all import *
>>> poly_ring1 = PolynomialRing(QQ, names=('gen1',)); (gen1,) = poly_ring1._first_ngens(1)
>>> poly_ring2 = PolynomialRing(QQ, names=('gen2',)); (gen2,) = poly_ring2._first_ngens(1)
>>> huge_ring = PolynomialRing(poly_ring1, names=('x',)); (x,) = huge_ring._first_ngens(1)

Sage Live

poly_ring1.<gen1> = PolynomialRing(QQ)
poly_ring2.<gen2> = PolynomialRing(QQ)
huge_ring.<x> = PolynomialRing(poly_ring1)

The generator of the first ring gets coerced in as itself, since it is the base ring.

Sage

sage: huge_ring(gen1)
gen1

Python

>>> from sage.all import *
>>> huge_ring(gen1)
gen1

Sage Live

huge_ring(gen1)

The generator of the second ring gets mapped via the natural map sending one generator to the other.

Sage

sage: huge_ring(gen2)
x

Python

>>> from sage.all import *
>>> huge_ring(gen2)
x

Sage Live

huge_ring(gen2)

With power series the behavior is the same.

Sage

sage: power_ring1.<gen1> = PowerSeriesRing(QQ)
sage: power_ring2.<gen2> = PowerSeriesRing(QQ)
sage: huge_power_ring.<x> = PowerSeriesRing(power_ring1)
sage: huge_power_ring(gen1)
gen1
sage: huge_power_ring(gen2)
x

Python

>>> from sage.all import *
>>> power_ring1 = PowerSeriesRing(QQ, names=('gen1',)); (gen1,) = power_ring1._first_ngens(1)
>>> power_ring2 = PowerSeriesRing(QQ, names=('gen2',)); (gen2,) = power_ring2._first_ngens(1)
>>> huge_power_ring = PowerSeriesRing(power_ring1, names=('x',)); (x,) = huge_power_ring._first_ngens(1)
>>> huge_power_ring(gen1)
gen1
>>> huge_power_ring(gen2)
x

Sage Live

power_ring1.<gen1> = PowerSeriesRing(QQ)
power_ring2.<gen2> = PowerSeriesRing(QQ)
huge_power_ring.<x> = PowerSeriesRing(power_ring1)
huge_power_ring(gen1)
huge_power_ring(gen2)

class sage.rings.power_series_ring_element.PowerSeries[source]¶

Bases: AlgebraElement

A power series. Base class of univariate and multivariate power series. The following methods are available with both types of objects.

O(prec)[source]¶

Return this series plus \(O(x^\text{prec})\). Does not change self.

EXAMPLES:

Sage

sage: R.<x> = PowerSeriesRing(ZZ)
sage: p = 1 + x^2 + x^10; p
1 + x^2 + x^10
sage: p.O(15)
1 + x^2 + x^10 + O(x^15)
sage: p.O(5)
1 + x^2 + O(x^5)
sage: p.O(-5)
Traceback (most recent call last):
...
ValueError: prec (= -5) must be nonnegative

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> p = Integer(1) + x**Integer(2) + x**Integer(10); p
1 + x^2 + x^10
>>> p.O(Integer(15))
1 + x^2 + x^10 + O(x^15)
>>> p.O(Integer(5))
1 + x^2 + O(x^5)
>>> p.O(-Integer(5))
Traceback (most recent call last):
...
ValueError: prec (= -5) must be nonnegative

Sage Live

R.<x> = PowerSeriesRing(ZZ)
p = 1 + x^2 + x^10; p
p.O(15)
p.O(5)
p.O(-5)

V(n)[source]¶

If \(f = \sum a_m x^m\), then this function returns \(\sum a_m x^{nm}\).

EXAMPLES:

Sage

sage: R.<x> = PowerSeriesRing(ZZ)
sage: p = 1 + x^2 + x^10; p
1 + x^2 + x^10
sage: p.V(3)
1 + x^6 + x^30
sage: (p + O(x^20)).V(3)
1 + x^6 + x^30 + O(x^60)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> p = Integer(1) + x**Integer(2) + x**Integer(10); p
1 + x^2 + x^10
>>> p.V(Integer(3))
1 + x^6 + x^30
>>> (p + O(x**Integer(20))).V(Integer(3))
1 + x^6 + x^30 + O(x^60)

Sage Live

R.<x> = PowerSeriesRing(ZZ)
p = 1 + x^2 + x^10; p
p.V(3)
(p + O(x^20)).V(3)

add_bigoh(prec)[source]¶

Return the power series of precision at most prec got by adding \(O(q^\text{prec})\) to \(f\), where \(q\) is the variable.

EXAMPLES:

Sage

sage: R.<A> = RDF[[]]
sage: f = (1+A+O(A^5))^5; f
1.0 + 5.0*A + 10.0*A^2 + 10.0*A^3 + 5.0*A^4 + O(A^5)
sage: f.add_bigoh(3)
1.0 + 5.0*A + 10.0*A^2 + O(A^3)
sage: f.add_bigoh(5)
1.0 + 5.0*A + 10.0*A^2 + 10.0*A^3 + 5.0*A^4 + O(A^5)

Python

>>> from sage.all import *
>>> R = RDF[['A']]; (A,) = R._first_ngens(1)
>>> f = (Integer(1)+A+O(A**Integer(5)))**Integer(5); f
1.0 + 5.0*A + 10.0*A^2 + 10.0*A^3 + 5.0*A^4 + O(A^5)
>>> f.add_bigoh(Integer(3))
1.0 + 5.0*A + 10.0*A^2 + O(A^3)
>>> f.add_bigoh(Integer(5))
1.0 + 5.0*A + 10.0*A^2 + 10.0*A^3 + 5.0*A^4 + O(A^5)

Sage Live

R.<A> = RDF[[]]
f = (1+A+O(A^5))^5; f
f.add_bigoh(3)
f.add_bigoh(5)

base_extend(R)[source]¶

Return a copy of this power series but with coefficients in R.

The following coercion uses base_extend implicitly:

Sage

sage: R.<t> = ZZ[['t']]
sage: (t - t^2) * Mod(1, 3)
t + 2*t^2

Python

>>> from sage.all import *
>>> R = ZZ[['t']]; (t,) = R._first_ngens(1)
>>> (t - t**Integer(2)) * Mod(Integer(1), Integer(3))
t + 2*t^2

Sage Live

R.<t> = ZZ[['t']]
(t - t^2) * Mod(1, 3)

base_ring()[source]¶

Return the base ring that this power series is defined over.

EXAMPLES:

Sage

sage: R.<t> = GF(49,'alpha')[[]]                                            # needs sage.rings.finite_rings
sage: (t^2 + O(t^3)).base_ring()                                            # needs sage.rings.finite_rings
Finite Field in alpha of size 7^2

Python

>>> from sage.all import *
>>> R = GF(Integer(49),'alpha')[['t']]; (t,) = R._first_ngens(1)# needs sage.rings.finite_rings
>>> (t**Integer(2) + O(t**Integer(3))).base_ring()                                            # needs sage.rings.finite_rings
Finite Field in alpha of size 7^2

Sage Live

R.<t> = GF(49,'alpha')[[]]                                            # needs sage.rings.finite_rings
(t^2 + O(t^3)).base_ring()                                            # needs sage.rings.finite_rings

change_ring(R)[source]¶

Change if possible the coefficients of self to lie in R.

EXAMPLES:

Sage

sage: R.<T> = QQ[[]]; R
Power Series Ring in T over Rational Field
sage: f = 1 - 1/2*T + 1/3*T^2 + O(T^3)
sage: f.base_extend(GF(5))
Traceback (most recent call last):
...
TypeError: no base extension defined
sage: f.change_ring(GF(5))
1 + 2*T + 2*T^2 + O(T^3)
sage: f.change_ring(GF(3))
Traceback (most recent call last):
...
ZeroDivisionError: inverse of Mod(0, 3) does not exist

Python

>>> from sage.all import *
>>> R = QQ[['T']]; (T,) = R._first_ngens(1); R
Power Series Ring in T over Rational Field
>>> f = Integer(1) - Integer(1)/Integer(2)*T + Integer(1)/Integer(3)*T**Integer(2) + O(T**Integer(3))
>>> f.base_extend(GF(Integer(5)))
Traceback (most recent call last):
...
TypeError: no base extension defined
>>> f.change_ring(GF(Integer(5)))
1 + 2*T + 2*T^2 + O(T^3)
>>> f.change_ring(GF(Integer(3)))
Traceback (most recent call last):
...
ZeroDivisionError: inverse of Mod(0, 3) does not exist

Sage Live

R.<T> = QQ[[]]; R
f = 1 - 1/2*T + 1/3*T^2 + O(T^3)
f.base_extend(GF(5))
f.change_ring(GF(5))
f.change_ring(GF(3))

We can only change the ring if there is a __call__ coercion defined. The following succeeds because ZZ(K(4)) is defined.

Sage

sage: K.<a> = NumberField(cyclotomic_polynomial(3), 'a')                    # needs sage.rings.number_field
sage: R.<t> = K[['t']]                                                      # needs sage.rings.number_field
sage: (4*t).change_ring(ZZ)                                                 # needs sage.rings.number_field
4*t

Python

>>> from sage.all import *
>>> K = NumberField(cyclotomic_polynomial(Integer(3)), 'a', names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> R = K[['t']]; (t,) = R._first_ngens(1)# needs sage.rings.number_field
>>> (Integer(4)*t).change_ring(ZZ)                                                 # needs sage.rings.number_field
4*t

Sage Live

K.<a> = NumberField(cyclotomic_polynomial(3), 'a')                    # needs sage.rings.number_field
R.<t> = K[['t']]                                                      # needs sage.rings.number_field
(4*t).change_ring(ZZ)                                                 # needs sage.rings.number_field

This does not succeed because ZZ(K(a+1)) is not defined.

Sage

sage: K.<a> = NumberField(cyclotomic_polynomial(3), 'a')                    # needs sage.rings.number_field
sage: R.<t> = K[['t']]                                                      # needs sage.rings.number_field
sage: ((a+1)*t).change_ring(ZZ)                                             # needs sage.rings.number_field
Traceback (most recent call last):
...
TypeError: Unable to coerce a + 1 to an integer

Python

>>> from sage.all import *
>>> K = NumberField(cyclotomic_polynomial(Integer(3)), 'a', names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> R = K[['t']]; (t,) = R._first_ngens(1)# needs sage.rings.number_field
>>> ((a+Integer(1))*t).change_ring(ZZ)                                             # needs sage.rings.number_field
Traceback (most recent call last):
...
TypeError: Unable to coerce a + 1 to an integer

Sage Live

K.<a> = NumberField(cyclotomic_polynomial(3), 'a')                    # needs sage.rings.number_field
R.<t> = K[['t']]                                                      # needs sage.rings.number_field
((a+1)*t).change_ring(ZZ)                                             # needs sage.rings.number_field

coefficients()[source]¶

Return the nonzero coefficients of self.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ)
sage: f = t + t^2 - 10/3*t^3
sage: f.coefficients()
[1, 1, -10/3]

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = t + t**Integer(2) - Integer(10)/Integer(3)*t**Integer(3)
>>> f.coefficients()
[1, 1, -10/3]

Sage Live

R.<t> = PowerSeriesRing(QQ)
f = t + t^2 - 10/3*t^3
f.coefficients()

common_prec(f)[source]¶

Return minimum precision of \(f\) and self.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t',)); (t,) = R._first_ngens(1)

Sage Live

R.<t> = PowerSeriesRing(QQ)

Sage

sage: f = t + t^2 + O(t^3)
sage: g = t + t^3 + t^4 + O(t^4)
sage: f.common_prec(g)
3
sage: g.common_prec(f)
3

Python

>>> from sage.all import *
>>> f = t + t**Integer(2) + O(t**Integer(3))
>>> g = t + t**Integer(3) + t**Integer(4) + O(t**Integer(4))
>>> f.common_prec(g)
3
>>> g.common_prec(f)
3

Sage Live

f = t + t^2 + O(t^3)
g = t + t^3 + t^4 + O(t^4)
f.common_prec(g)
g.common_prec(f)

Python

>>> from sage.all import *
>>> f = t + t**Integer(2) + O(t**Integer(3))
>>> g = t + t**Integer(3) + t**Integer(4) + O(t**Integer(4))
>>> f.common_prec(g)
3
>>> g.common_prec(f)
3

Sage Live

f = t + t^2 + O(t^3)
g = t + t^3 + t^4 + O(t^4)
f.common_prec(g)
g.common_prec(f)

Sage

sage: f = t + t^2 + O(t^3)
sage: g = t^2
sage: f.common_prec(g)
3
sage: g.common_prec(f)
3

Python

>>> from sage.all import *
>>> f = t + t**Integer(2) + O(t**Integer(3))
>>> g = t**Integer(2)
>>> f.common_prec(g)
3
>>> g.common_prec(f)
3

Sage Live

f = t + t^2 + O(t^3)
g = t^2
f.common_prec(g)
g.common_prec(f)

Python

>>> from sage.all import *
>>> f = t + t**Integer(2) + O(t**Integer(3))
>>> g = t**Integer(2)
>>> f.common_prec(g)
3
>>> g.common_prec(f)
3

Sage Live

f = t + t^2 + O(t^3)
g = t^2
f.common_prec(g)
g.common_prec(f)

Sage

sage: f = t + t^2
sage: f = t^2
sage: f.common_prec(g)
+Infinity

Python

>>> from sage.all import *
>>> f = t + t**Integer(2)
>>> f = t**Integer(2)
>>> f.common_prec(g)
+Infinity

Sage Live

f = t + t^2
f = t^2
f.common_prec(g)

Python

>>> from sage.all import *
>>> f = t + t**Integer(2)
>>> f = t**Integer(2)
>>> f.common_prec(g)
+Infinity

Sage Live

f = t + t^2
f = t^2
f.common_prec(g)

cos(prec='infinity')[source]¶

Apply cos to the formal power series.

INPUT:

prec – integer or infinity; the degree to truncate the result to

OUTPUT: a new power series

EXAMPLES:

For one variable:

Sage

sage: t = PowerSeriesRing(QQ, 't').gen()
sage: f = (t + t**2).O(4)
sage: cos(f)
1 - 1/2*t^2 - t^3 + O(t^4)

Python

>>> from sage.all import *
>>> t = PowerSeriesRing(QQ, 't').gen()
>>> f = (t + t**Integer(2)).O(Integer(4))
>>> cos(f)
1 - 1/2*t^2 - t^3 + O(t^4)

Sage Live

t = PowerSeriesRing(QQ, 't').gen()
f = (t + t**2).O(4)
cos(f)

For several variables:

Sage

sage: T.<a,b> = PowerSeriesRing(ZZ,2)
sage: f = a + b + a*b + T.O(3)
sage: cos(f)
1 - 1/2*a^2 - a*b - 1/2*b^2 + O(a, b)^3
sage: f.cos()
1 - 1/2*a^2 - a*b - 1/2*b^2 + O(a, b)^3
sage: f.cos(prec=2)
1 + O(a, b)^2

Python

>>> from sage.all import *
>>> T = PowerSeriesRing(ZZ,Integer(2), names=('a', 'b',)); (a, b,) = T._first_ngens(2)
>>> f = a + b + a*b + T.O(Integer(3))
>>> cos(f)
1 - 1/2*a^2 - a*b - 1/2*b^2 + O(a, b)^3
>>> f.cos()
1 - 1/2*a^2 - a*b - 1/2*b^2 + O(a, b)^3
>>> f.cos(prec=Integer(2))
1 + O(a, b)^2

Sage Live

T.<a,b> = PowerSeriesRing(ZZ,2)
f = a + b + a*b + T.O(3)
cos(f)
f.cos()
f.cos(prec=2)

If the power series has a nonzero constant coefficient \(c\), one raises an error:

Sage

sage: g = 2+f
sage: cos(g)
Traceback (most recent call last):
...
ValueError: can only apply cos to formal power series with zero constant term

Python

>>> from sage.all import *
>>> g = Integer(2)+f
>>> cos(g)
Traceback (most recent call last):
...
ValueError: can only apply cos to formal power series with zero constant term

Sage Live

g = 2+f
cos(g)

If no precision is specified, the default precision is used:

Sage

sage: T.default_prec()
12
sage: cos(a)
1 - 1/2*a^2 + 1/24*a^4 - 1/720*a^6 + 1/40320*a^8 - 1/3628800*a^10 + O(a, b)^12
sage: a.cos(prec=5)
1 - 1/2*a^2 + 1/24*a^4 + O(a, b)^5
sage: cos(a + T.O(5))
1 - 1/2*a^2 + 1/24*a^4 + O(a, b)^5

Python

>>> from sage.all import *
>>> T.default_prec()
12
>>> cos(a)
1 - 1/2*a^2 + 1/24*a^4 - 1/720*a^6 + 1/40320*a^8 - 1/3628800*a^10 + O(a, b)^12
>>> a.cos(prec=Integer(5))
1 - 1/2*a^2 + 1/24*a^4 + O(a, b)^5
>>> cos(a + T.O(Integer(5)))
1 - 1/2*a^2 + 1/24*a^4 + O(a, b)^5

Sage Live

T.default_prec()
cos(a)
a.cos(prec=5)
cos(a + T.O(5))

cosh(prec='infinity')[source]¶

Apply cosh to the formal power series.

INPUT:

prec – integer or infinity; the degree to truncate the result to

OUTPUT: a new power series

EXAMPLES:

For one variable:

Sage

sage: t = PowerSeriesRing(QQ, 't').gen()
sage: f = (t + t**2).O(4)
sage: cosh(f)
1 + 1/2*t^2 + t^3 + O(t^4)

Python

>>> from sage.all import *
>>> t = PowerSeriesRing(QQ, 't').gen()
>>> f = (t + t**Integer(2)).O(Integer(4))
>>> cosh(f)
1 + 1/2*t^2 + t^3 + O(t^4)

Sage Live

t = PowerSeriesRing(QQ, 't').gen()
f = (t + t**2).O(4)
cosh(f)

For several variables:

Sage

sage: T.<a,b> = PowerSeriesRing(ZZ,2)
sage: f = a + b + a*b + T.O(3)
sage: cosh(f)
1 + 1/2*a^2 + a*b + 1/2*b^2 + O(a, b)^3
sage: f.cosh()
1 + 1/2*a^2 + a*b + 1/2*b^2 + O(a, b)^3
sage: f.cosh(prec=2)
1 + O(a, b)^2

Python

>>> from sage.all import *
>>> T = PowerSeriesRing(ZZ,Integer(2), names=('a', 'b',)); (a, b,) = T._first_ngens(2)
>>> f = a + b + a*b + T.O(Integer(3))
>>> cosh(f)
1 + 1/2*a^2 + a*b + 1/2*b^2 + O(a, b)^3
>>> f.cosh()
1 + 1/2*a^2 + a*b + 1/2*b^2 + O(a, b)^3
>>> f.cosh(prec=Integer(2))
1 + O(a, b)^2

Sage Live

T.<a,b> = PowerSeriesRing(ZZ,2)
f = a + b + a*b + T.O(3)
cosh(f)
f.cosh()
f.cosh(prec=2)

If the power series has a nonzero constant coefficient \(c\), one raises an error:

Sage

sage: g = 2 + f
sage: cosh(g)
Traceback (most recent call last):
...
ValueError: can only apply cosh to formal power series with zero
constant term

Python

>>> from sage.all import *
>>> g = Integer(2) + f
>>> cosh(g)
Traceback (most recent call last):
...
ValueError: can only apply cosh to formal power series with zero
constant term

Sage Live

g = 2 + f
cosh(g)

If no precision is specified, the default precision is used:

Sage

sage: T.default_prec()
12
sage: cosh(a)
1 + 1/2*a^2 + 1/24*a^4 + 1/720*a^6 + 1/40320*a^8 + 1/3628800*a^10 +
 O(a, b)^12
sage: a.cosh(prec=5)
1 + 1/2*a^2 + 1/24*a^4 + O(a, b)^5
sage: cosh(a + T.O(5))
1 + 1/2*a^2 + 1/24*a^4 + O(a, b)^5

Python

>>> from sage.all import *
>>> T.default_prec()
12
>>> cosh(a)
1 + 1/2*a^2 + 1/24*a^4 + 1/720*a^6 + 1/40320*a^8 + 1/3628800*a^10 +
 O(a, b)^12
>>> a.cosh(prec=Integer(5))
1 + 1/2*a^2 + 1/24*a^4 + O(a, b)^5
>>> cosh(a + T.O(Integer(5)))
1 + 1/2*a^2 + 1/24*a^4 + O(a, b)^5

Sage Live

T.default_prec()
cosh(a)
a.cosh(prec=5)
cosh(a + T.O(5))

degree()[source]¶

Return the degree of this power series, which is by definition the degree of the underlying polynomial.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ, sparse=True)
sage: f = t^100000 + O(t^10000000)
sage: f.degree()
100000

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, sparse=True, names=('t',)); (t,) = R._first_ngens(1)
>>> f = t**Integer(100000) + O(t**Integer(10000000))
>>> f.degree()
100000

Sage Live

R.<t> = PowerSeriesRing(QQ, sparse=True)
f = t^100000 + O(t^10000000)
f.degree()

derivative(*args)[source]¶

The formal derivative of this power series, with respect to variables supplied in args.

Multiple variables and iteration counts may be supplied; see documentation for the global derivative() function for more details.

See also

ogf_to_egf() for the inverse method.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ)
sage: f = t + t^2/factorial(2) + 2*t^3/factorial(3)
sage: f.egf_to_ogf()
t + t^2 + 2*t^3

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = t + t**Integer(2)/factorial(Integer(2)) + Integer(2)*t**Integer(3)/factorial(Integer(3))
>>> f.egf_to_ogf()
t + t^2 + 2*t^3

Sage Live

R.<t> = PowerSeriesRing(QQ)
f = t + t^2/factorial(2) + 2*t^3/factorial(3)
f.egf_to_ogf()

exp(prec=None)[source]¶

Return exp of this power series to the indicated precision.

INPUT:

prec – integer (default: self.parent().default_prec)

ALGORITHM: See solve_linear_de().

Note

Screwy things can happen if the coefficient ring is not a field of characteristic zero. See solve_linear_de().

AUTHORS:

David Harvey (2006-09-08): rewrote to use simplest possible “lazy” algorithm.
David Harvey (2006-09-10): rewrote to use divide-and-conquer strategy.
David Harvey (2006-09-11): factored functionality out to solve_linear_de().
Sourav Sen Gupta, David Harvey (2008-11): handle constant term

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ, default_prec=10)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, default_prec=Integer(10), names=('t',)); (t,) = R._first_ngens(1)

Sage Live

R.<t> = PowerSeriesRing(QQ, default_prec=10)

Check that \(\exp(t)\) is, well, \(\exp(t)\):

Sage

sage: (t + O(t^10)).exp()
1 + t + 1/2*t^2 + 1/6*t^3 + 1/24*t^4 + 1/120*t^5 + 1/720*t^6
  + 1/5040*t^7 + 1/40320*t^8 + 1/362880*t^9 + O(t^10)

Python

>>> from sage.all import *
>>> (t + O(t**Integer(10))).exp()
1 + t + 1/2*t^2 + 1/6*t^3 + 1/24*t^4 + 1/120*t^5 + 1/720*t^6
  + 1/5040*t^7 + 1/40320*t^8 + 1/362880*t^9 + O(t^10)

Sage Live

(t + O(t^10)).exp()

Check that \(\exp(\log(1+t))\) is \(1+t\):

Sage

sage: (sum([-(-t)^n/n for n in range(1, 10)]) + O(t^10)).exp()
1 + t + O(t^10)

Python

>>> from sage.all import *
>>> (sum([-(-t)**n/n for n in range(Integer(1), Integer(10))]) + O(t**Integer(10))).exp()
1 + t + O(t^10)

Sage Live

(sum([-(-t)^n/n for n in range(1, 10)]) + O(t^10)).exp()

Check that \(\exp(2t + t^2 - t^5)\) is whatever it is:

Sage

sage: (2*t + t^2 - t^5 + O(t^10)).exp()
1 + 2*t + 3*t^2 + 10/3*t^3 + 19/6*t^4 + 8/5*t^5 - 7/90*t^6
  - 538/315*t^7 - 425/168*t^8 - 30629/11340*t^9 + O(t^10)

Python

>>> from sage.all import *
>>> (Integer(2)*t + t**Integer(2) - t**Integer(5) + O(t**Integer(10))).exp()
1 + 2*t + 3*t^2 + 10/3*t^3 + 19/6*t^4 + 8/5*t^5 - 7/90*t^6
  - 538/315*t^7 - 425/168*t^8 - 30629/11340*t^9 + O(t^10)

Sage Live

(2*t + t^2 - t^5 + O(t^10)).exp()

Check requesting lower precision:

Sage

sage: (t + t^2 - t^5 + O(t^10)).exp(5)
1 + t + 3/2*t^2 + 7/6*t^3 + 25/24*t^4 + O(t^5)

Python

>>> from sage.all import *
>>> (t + t**Integer(2) - t**Integer(5) + O(t**Integer(10))).exp(Integer(5))
1 + t + 3/2*t^2 + 7/6*t^3 + 25/24*t^4 + O(t^5)

Sage Live

(t + t^2 - t^5 + O(t^10)).exp(5)

Can’t get more precision than the input:

Sage

sage: (t + t^2 + O(t^3)).exp(10)
1 + t + 3/2*t^2 + O(t^3)

Python

>>> from sage.all import *
>>> (t + t**Integer(2) + O(t**Integer(3))).exp(Integer(10))
1 + t + 3/2*t^2 + O(t^3)

Sage Live

(t + t^2 + O(t^3)).exp(10)

Check some boundary cases:

Sage

sage: (t + O(t^2)).exp(1)
1 + O(t)
sage: (t + O(t^2)).exp(0)
O(t^0)

Python

>>> from sage.all import *
>>> (t + O(t**Integer(2))).exp(Integer(1))
1 + O(t)
>>> (t + O(t**Integer(2))).exp(Integer(0))
O(t^0)

Sage Live

(t + O(t^2)).exp(1)
(t + O(t^2)).exp(0)

Handle nonzero constant term (fixes Issue #4477):

Sage

sage: # needs sage.rings.real_mpfr
sage: R.<x> = PowerSeriesRing(RR)
sage: (1 + x + x^2 + O(x^3)).exp()
2.71828182845905 + 2.71828182845905*x + 4.07742274268857*x^2 + O(x^3)

Python

>>> from sage.all import *
>>> # needs sage.rings.real_mpfr
>>> R = PowerSeriesRing(RR, names=('x',)); (x,) = R._first_ngens(1)
>>> (Integer(1) + x + x**Integer(2) + O(x**Integer(3))).exp()
2.71828182845905 + 2.71828182845905*x + 4.07742274268857*x^2 + O(x^3)

Sage Live

# needs sage.rings.real_mpfr
R.<x> = PowerSeriesRing(RR)
(1 + x + x^2 + O(x^3)).exp()

Sage

sage: R.<x> = PowerSeriesRing(ZZ)
sage: (1 + x + O(x^2)).exp()                                                # needs sage.symbolic
Traceback (most recent call last):
...
ArithmeticError: exponential of constant term does not belong
to coefficient ring (consider working in a larger ring)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> (Integer(1) + x + O(x**Integer(2))).exp()                                                # needs sage.symbolic
Traceback (most recent call last):
...
ArithmeticError: exponential of constant term does not belong
to coefficient ring (consider working in a larger ring)

Sage Live

R.<x> = PowerSeriesRing(ZZ)
(1 + x + O(x^2)).exp()                                                # needs sage.symbolic

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> (Integer(1) + x + O(x**Integer(2))).exp()                                                # needs sage.symbolic
Traceback (most recent call last):
...
ArithmeticError: exponential of constant term does not belong
to coefficient ring (consider working in a larger ring)

Sage Live

R.<x> = PowerSeriesRing(ZZ)
(1 + x + O(x^2)).exp()                                                # needs sage.symbolic

Sage

sage: R.<x> = PowerSeriesRing(GF(5))
sage: (1 + x + O(x^2)).exp()
Traceback (most recent call last):
...
ArithmeticError: constant term of power series does not support exponentiation

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(GF(Integer(5)), names=('x',)); (x,) = R._first_ngens(1)
>>> (Integer(1) + x + O(x**Integer(2))).exp()
Traceback (most recent call last):
...
ArithmeticError: constant term of power series does not support exponentiation

Sage Live

R.<x> = PowerSeriesRing(GF(5))
(1 + x + O(x^2)).exp()

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(GF(Integer(5)), names=('x',)); (x,) = R._first_ngens(1)
>>> (Integer(1) + x + O(x**Integer(2))).exp()
Traceback (most recent call last):
...
ArithmeticError: constant term of power series does not support exponentiation

Sage Live

R.<x> = PowerSeriesRing(GF(5))
(1 + x + O(x^2)).exp()

exponents()[source]¶

Return the exponents appearing in self with nonzero coefficients.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ)
sage: f = t + t^2 - 10/3*t^3
sage: f.exponents()
[1, 2, 3]

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = t + t**Integer(2) - Integer(10)/Integer(3)*t**Integer(3)
>>> f.exponents()
[1, 2, 3]

Sage Live

R.<t> = PowerSeriesRing(QQ)
f = t + t^2 - 10/3*t^3
f.exponents()

inverse()[source]¶

Return the inverse of self, i.e., self^(-1).

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ, sparse=True)
sage: t.inverse()
t^-1
sage: type(_)
<class 'sage.rings.laurent_series_ring_element.LaurentSeries'>
sage: (1-t).inverse()
1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + ...

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, sparse=True, names=('t',)); (t,) = R._first_ngens(1)
>>> t.inverse()
t^-1
>>> type(_)
<class 'sage.rings.laurent_series_ring_element.LaurentSeries'>
>>> (Integer(1)-t).inverse()
1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + ...

Sage Live

R.<t> = PowerSeriesRing(QQ, sparse=True)
t.inverse()
type(_)
(1-t).inverse()

is_dense()[source]¶

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(ZZ)
sage: t.is_dense()
True
sage: R.<t> = PowerSeriesRing(ZZ, sparse=True)
sage: t.is_dense()
False

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('t',)); (t,) = R._first_ngens(1)
>>> t.is_dense()
True
>>> R = PowerSeriesRing(ZZ, sparse=True, names=('t',)); (t,) = R._first_ngens(1)
>>> t.is_dense()
False

Sage Live

R.<t> = PowerSeriesRing(ZZ)
t.is_dense()
R.<t> = PowerSeriesRing(ZZ, sparse=True)
t.is_dense()

is_gen()[source]¶

Return True if this is the generator (the variable) of the power series ring.

EXAMPLES:

Sage

sage: R.<t> = QQ[[]]
sage: t.is_gen()
True
sage: (1 + 2*t).is_gen()
False

Python

>>> from sage.all import *
>>> R = QQ[['t']]; (t,) = R._first_ngens(1)
>>> t.is_gen()
True
>>> (Integer(1) + Integer(2)*t).is_gen()
False

Sage Live

R.<t> = QQ[[]]
t.is_gen()
(1 + 2*t).is_gen()

Note that this only returns True on the actual generator, not on something that happens to be equal to it.

Sage

sage: (1*t).is_gen()
False
sage: 1*t == t
True

Python

>>> from sage.all import *
>>> (Integer(1)*t).is_gen()
False
>>> Integer(1)*t == t
True

Sage Live

(1*t).is_gen()
1*t == t

is_monomial()[source]¶

Return True if this element is a monomial. That is, if self is \(x^n\) for some nonnegative integer \(n\).

EXAMPLES:

Sage

sage: k.<z> = PowerSeriesRing(QQ, 'z')
sage: z.is_monomial()
True
sage: k(1).is_monomial()
True
sage: (z+1).is_monomial()
False
sage: (z^2909).is_monomial()
True
sage: (3*z^2909).is_monomial()
False

Python

>>> from sage.all import *
>>> k = PowerSeriesRing(QQ, 'z', names=('z',)); (z,) = k._first_ngens(1)
>>> z.is_monomial()
True
>>> k(Integer(1)).is_monomial()
True
>>> (z+Integer(1)).is_monomial()
False
>>> (z**Integer(2909)).is_monomial()
True
>>> (Integer(3)*z**Integer(2909)).is_monomial()
False

Sage Live

k.<z> = PowerSeriesRing(QQ, 'z')
z.is_monomial()
k(1).is_monomial()
(z+1).is_monomial()
(z^2909).is_monomial()
(3*z^2909).is_monomial()

is_sparse()[source]¶

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(ZZ)
sage: t.is_sparse()
False
sage: R.<t> = PowerSeriesRing(ZZ, sparse=True)
sage: t.is_sparse()
True

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('t',)); (t,) = R._first_ngens(1)
>>> t.is_sparse()
False
>>> R = PowerSeriesRing(ZZ, sparse=True, names=('t',)); (t,) = R._first_ngens(1)
>>> t.is_sparse()
True

Sage Live

R.<t> = PowerSeriesRing(ZZ)
t.is_sparse()
R.<t> = PowerSeriesRing(ZZ, sparse=True)
t.is_sparse()

is_square()[source]¶

Return True if this function has a square root in this ring, e.g., there is an element \(y\) in self.parent() such that \(y^2\) equals self.

ALGORITHM: If the base ring is a field, this is true whenever the power series has even valuation and the leading coefficient is a perfect square.

For an integral domain, it attempts the square root in the fraction field and tests whether or not the result lies in the original ring.

EXAMPLES:

Sage

sage: K.<t> = PowerSeriesRing(QQ, 't', 5)
sage: (1+t).is_square()
True
sage: (2+t).is_square()
False
sage: (2+t.change_ring(RR)).is_square()
True
sage: t.is_square()
False
sage: K.<t> = PowerSeriesRing(ZZ, 't', 5)
sage: (1+t).is_square()
False
sage: f = (1+t)^100
sage: f.is_square()
True

Python

>>> from sage.all import *
>>> K = PowerSeriesRing(QQ, 't', Integer(5), names=('t',)); (t,) = K._first_ngens(1)
>>> (Integer(1)+t).is_square()
True
>>> (Integer(2)+t).is_square()
False
>>> (Integer(2)+t.change_ring(RR)).is_square()
True
>>> t.is_square()
False
>>> K = PowerSeriesRing(ZZ, 't', Integer(5), names=('t',)); (t,) = K._first_ngens(1)
>>> (Integer(1)+t).is_square()
False
>>> f = (Integer(1)+t)**Integer(100)
>>> f.is_square()
True

Sage Live

K.<t> = PowerSeriesRing(QQ, 't', 5)
(1+t).is_square()
(2+t).is_square()
(2+t.change_ring(RR)).is_square()
t.is_square()
K.<t> = PowerSeriesRing(ZZ, 't', 5)
(1+t).is_square()
f = (1+t)^100
f.is_square()

is_unit()[source]¶

Return True if this power series is invertible.

A power series is invertible precisely when the constant term is invertible.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(ZZ)
sage: (-1 + t - t^5).is_unit()
True
sage: (3 + t - t^5).is_unit()
False
sage: O(t^0).is_unit()
False

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('t',)); (t,) = R._first_ngens(1)
>>> (-Integer(1) + t - t**Integer(5)).is_unit()
True
>>> (Integer(3) + t - t**Integer(5)).is_unit()
False
>>> O(t**Integer(0)).is_unit()
False

Sage Live

R.<t> = PowerSeriesRing(ZZ)
(-1 + t - t^5).is_unit()
(3 + t - t^5).is_unit()
O(t^0).is_unit()

AUTHORS:

David Harvey (2006-09-03)

jacobi_continued_fraction()[source]¶

Return the Jacobi continued fraction of self.

The J-fraction or Jacobi continued fraction of a power series is a continued fraction expansion with steps of size two. We use the following convention

\[1 / (1 + A_0 t + B_0 t^2 / (1 + A_1 t + B_1 t^2 / (1 + \cdots)))\]

OUTPUT:

tuple of pairs \((A_n, B_n)\) for \(n \geq 0\)

The expansion is done as long as possible given the precision. Whenever the expansion is not well-defined, because it would require to divide by zero, an exception is raised.

See section 2.7 of [Kra1999det] for the close relationship of this kind of expansion with Hankel determinants and orthogonal polynomials.

EXAMPLES:

Sage

sage: t = PowerSeriesRing(QQ, 't').gen()
sage: s = sum(factorial(k) * t**k for k in range(12)).O(12)
sage: s.jacobi_continued_fraction()
((-1, -1), (-3, -4), (-5, -9), (-7, -16), (-9, -25))

Python

>>> from sage.all import *
>>> t = PowerSeriesRing(QQ, 't').gen()
>>> s = sum(factorial(k) * t**k for k in range(Integer(12))).O(Integer(12))
>>> s.jacobi_continued_fraction()
((-1, -1), (-3, -4), (-5, -9), (-7, -16), (-9, -25))

Sage Live

t = PowerSeriesRing(QQ, 't').gen()
s = sum(factorial(k) * t**k for k in range(12)).O(12)
s.jacobi_continued_fraction()

Another example:

Sage

sage: (log(1+t)/t).jacobi_continued_fraction()
((1/2, -1/12),
 (1/2, -1/15),
 (1/2, -9/140),
 (1/2, -4/63),
 (1/2, -25/396),
 (1/2, -9/143),
 (1/2, -49/780),
 (1/2, -16/255),
 (1/2, -81/1292))

Python

>>> from sage.all import *
>>> (log(Integer(1)+t)/t).jacobi_continued_fraction()
((1/2, -1/12),
 (1/2, -1/15),
 (1/2, -9/140),
 (1/2, -4/63),
 (1/2, -25/396),
 (1/2, -9/143),
 (1/2, -49/780),
 (1/2, -16/255),
 (1/2, -81/1292))

Sage Live

(log(1+t)/t).jacobi_continued_fraction()

laurent_series()[source]¶

Return the Laurent series associated to this power series, i.e., this series considered as a Laurent series.

EXAMPLES:

Sage

sage: k.<w> = QQ[[]]
sage: f = 1 + 17*w + 15*w^3 + O(w^5)
sage: parent(f)
Power Series Ring in w over Rational Field
sage: g = f.laurent_series(); g
1 + 17*w + 15*w^3 + O(w^5)

Python

>>> from sage.all import *
>>> k = QQ[['w']]; (w,) = k._first_ngens(1)
>>> f = Integer(1) + Integer(17)*w + Integer(15)*w**Integer(3) + O(w**Integer(5))
>>> parent(f)
Power Series Ring in w over Rational Field
>>> g = f.laurent_series(); g
1 + 17*w + 15*w^3 + O(w^5)

Sage Live

k.<w> = QQ[[]]
f = 1 + 17*w + 15*w^3 + O(w^5)
parent(f)
g = f.laurent_series(); g

lift_to_precision(absprec=None)[source]¶

Return a congruent power series with absolute precision at least absprec.

INPUT:

absprec – integer or None (default: None); the absolute precision of the result. If None, lifts to an exact element.

EXAMPLES:

Sage

sage: A.<t> = PowerSeriesRing(GF(5))
sage: x = t + t^2 + O(t^5)
sage: x.lift_to_precision(10)
t + t^2 + O(t^10)
sage: x.lift_to_precision()
t + t^2

Python

>>> from sage.all import *
>>> A = PowerSeriesRing(GF(Integer(5)), names=('t',)); (t,) = A._first_ngens(1)
>>> x = t + t**Integer(2) + O(t**Integer(5))
>>> x.lift_to_precision(Integer(10))
t + t^2 + O(t^10)
>>> x.lift_to_precision()
t + t^2

Sage Live

A.<t> = PowerSeriesRing(GF(5))
x = t + t^2 + O(t^5)
x.lift_to_precision(10)
x.lift_to_precision()

list()[source]¶

See this method in derived classes:

Implementations MUST override this in the derived class.

EXAMPLES:

Sage

sage: from sage.rings.power_series_ring_element import PowerSeries
sage: R.<x> = PowerSeriesRing(ZZ)
sage: PowerSeries.list(1 + x^2)
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> from sage.rings.power_series_ring_element import PowerSeries
>>> R = PowerSeriesRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> PowerSeries.list(Integer(1) + x**Integer(2))
Traceback (most recent call last):
...
NotImplementedError

Sage Live

from sage.rings.power_series_ring_element import PowerSeries
R.<x> = PowerSeriesRing(ZZ)
PowerSeries.list(1 + x^2)

log(prec=None)[source]¶

Return log of this power series to the indicated precision.

This works only if the constant term of the power series is 1 or the base ring can take the logarithm of the constant coefficient.

INPUT:

prec – integer (default: self.parent().default_prec())

ALGORITHM: See solve_linear_de().

Warning

Screwy things can happen if the coefficient ring is not a field of characteristic zero. See solve_linear_de().

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ, default_prec=10)
sage: (1 + t + O(t^10)).log()
t - 1/2*t^2 + 1/3*t^3 - 1/4*t^4 + 1/5*t^5 - 1/6*t^6 + 1/7*t^7
  - 1/8*t^8 + 1/9*t^9 + O(t^10)

sage: t.exp().log()
t + O(t^10)

sage: (1 + t).log().exp()
1 + t + O(t^10)

sage: (-1 + t + O(t^10)).log()
Traceback (most recent call last):
...
ArithmeticError: constant term of power series is not 1

sage: # needs sage.rings.real_mpfr
sage: R.<t> = PowerSeriesRing(RR)
sage: (2 + t).log().exp()
2.00000000000000 + 1.00000000000000*t + O(t^20)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, default_prec=Integer(10), names=('t',)); (t,) = R._first_ngens(1)
>>> (Integer(1) + t + O(t**Integer(10))).log()
t - 1/2*t^2 + 1/3*t^3 - 1/4*t^4 + 1/5*t^5 - 1/6*t^6 + 1/7*t^7
  - 1/8*t^8 + 1/9*t^9 + O(t^10)

>>> t.exp().log()
t + O(t^10)

>>> (Integer(1) + t).log().exp()
1 + t + O(t^10)

>>> (-Integer(1) + t + O(t**Integer(10))).log()
Traceback (most recent call last):
...
ArithmeticError: constant term of power series is not 1

>>> # needs sage.rings.real_mpfr
>>> R = PowerSeriesRing(RR, names=('t',)); (t,) = R._first_ngens(1)
>>> (Integer(2) + t).log().exp()
2.00000000000000 + 1.00000000000000*t + O(t^20)

Sage Live

R.<t> = PowerSeriesRing(QQ, default_prec=10)
(1 + t + O(t^10)).log()
t.exp().log()
(1 + t).log().exp()
(-1 + t + O(t^10)).log()
# needs sage.rings.real_mpfr
R.<t> = PowerSeriesRing(RR)
(2 + t).log().exp()

map_coefficients(f, new_base_ring=None)[source]¶

Return the series obtained by applying f to the nonzero coefficients of self.

If f is a sage.categories.map.Map, then the resulting series will be defined over the codomain of f. Otherwise, the resulting polynomial will be over the same ring as self. Set new_base_ring to override this behaviour.

INPUT:

f – a callable that will be applied to the coefficients of self
new_base_ring – (optional) if given, the resulting polynomial will be defined over this ring

EXAMPLES:

Sage

sage: R.<x> = SR[[]]                                                        # needs sage.symbolic
sage: f = (1+I)*x^2 + 3*x - I                                               # needs sage.symbolic
sage: f.map_coefficients(lambda z: z.conjugate())                           # needs sage.symbolic
I + 3*x + (-I + 1)*x^2
sage: R.<x> = ZZ[[]]
sage: f = x^2 + 2
sage: f.map_coefficients(lambda a: a + 42)
44 + 43*x^2

Python

>>> from sage.all import *
>>> R = SR[['x']]; (x,) = R._first_ngens(1)# needs sage.symbolic
>>> f = (Integer(1)+I)*x**Integer(2) + Integer(3)*x - I                                               # needs sage.symbolic
>>> f.map_coefficients(lambda z: z.conjugate())                           # needs sage.symbolic
I + 3*x + (-I + 1)*x^2
>>> R = ZZ[['x']]; (x,) = R._first_ngens(1)
>>> f = x**Integer(2) + Integer(2)
>>> f.map_coefficients(lambda a: a + Integer(42))
44 + 43*x^2

Sage Live

R.<x> = SR[[]]                                                        # needs sage.symbolic
f = (1+I)*x^2 + 3*x - I                                               # needs sage.symbolic
f.map_coefficients(lambda z: z.conjugate())                           # needs sage.symbolic
R.<x> = ZZ[[]]
f = x^2 + 2
f.map_coefficients(lambda a: a + 42)

Examples with different base ring:

Sage

sage: R.<x> = ZZ[[]]
sage: k = GF(2)
sage: residue = lambda x: k(x)
sage: f = 4*x^2+x+3
sage: g = f.map_coefficients(residue); g
1 + x
sage: g.parent()
Power Series Ring in x over Integer Ring
sage: g = f.map_coefficients(residue, new_base_ring=k); g
1 + x
sage: g.parent()
Power Series Ring in x over Finite Field of size 2
sage: residue = k.coerce_map_from(ZZ)
sage: g = f.map_coefficients(residue); g
1 + x
sage: g.parent()
Power Series Ring in x over Finite Field of size 2

Python

>>> from sage.all import *
>>> R = ZZ[['x']]; (x,) = R._first_ngens(1)
>>> k = GF(Integer(2))
>>> residue = lambda x: k(x)
>>> f = Integer(4)*x**Integer(2)+x+Integer(3)
>>> g = f.map_coefficients(residue); g
1 + x
>>> g.parent()
Power Series Ring in x over Integer Ring
>>> g = f.map_coefficients(residue, new_base_ring=k); g
1 + x
>>> g.parent()
Power Series Ring in x over Finite Field of size 2
>>> residue = k.coerce_map_from(ZZ)
>>> g = f.map_coefficients(residue); g
1 + x
>>> g.parent()
Power Series Ring in x over Finite Field of size 2

Sage Live

R.<x> = ZZ[[]]
k = GF(2)
residue = lambda x: k(x)
f = 4*x^2+x+3
g = f.map_coefficients(residue); g
g.parent()
g = f.map_coefficients(residue, new_base_ring=k); g
g.parent()
residue = k.coerce_map_from(ZZ)
g = f.map_coefficients(residue); g
g.parent()

Tests other implementations:

Sage

sage: # needs sage.libs.pari
sage: R.<q> = PowerSeriesRing(GF(11), implementation='pari')
sage: f = q - q^3 + O(q^10)
sage: f.map_coefficients(lambda c: c - 2)
10*q + 8*q^3 + O(q^10)

Python

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> R = PowerSeriesRing(GF(Integer(11)), implementation='pari', names=('q',)); (q,) = R._first_ngens(1)
>>> f = q - q**Integer(3) + O(q**Integer(10))
>>> f.map_coefficients(lambda c: c - Integer(2))
10*q + 8*q^3 + O(q^10)

Sage Live

# needs sage.libs.pari
R.<q> = PowerSeriesRing(GF(11), implementation='pari')
f = q - q^3 + O(q^10)
f.map_coefficients(lambda c: c - 2)

nth_root(n, prec=None)[source]¶

Return the n-th root of this power series.

INPUT:

n – integer
prec – integer (optional); precision of the result. Though, if this series has finite precision, then the result cannot have larger precision.

EXAMPLES:

Sage

sage: R.<x> = QQ[[]]
sage: (1+x).nth_root(5)
1 + 1/5*x - 2/25*x^2 + ... + 12039376311816/2384185791015625*x^19 + O(x^20)

sage: (1 + x + O(x^5)).nth_root(5)
1 + 1/5*x - 2/25*x^2 + 6/125*x^3 - 21/625*x^4 + O(x^5)

Python

>>> from sage.all import *
>>> R = QQ[['x']]; (x,) = R._first_ngens(1)
>>> (Integer(1)+x).nth_root(Integer(5))
1 + 1/5*x - 2/25*x^2 + ... + 12039376311816/2384185791015625*x^19 + O(x^20)

>>> (Integer(1) + x + O(x**Integer(5))).nth_root(Integer(5))
1 + 1/5*x - 2/25*x^2 + 6/125*x^3 - 21/625*x^4 + O(x^5)

Sage Live

R.<x> = QQ[[]]
(1+x).nth_root(5)
(1 + x + O(x^5)).nth_root(5)

Check that the results are consistent with taking log and exponential:

Sage

sage: R.<x> = PowerSeriesRing(QQ, default_prec=100)
sage: p = (1 + 2*x - x^4)**200
sage: p1 = p.nth_root(1000, prec=100)
sage: p2 = (p.log()/1000).exp()
sage: p1.prec() == p2.prec() == 100
True
sage: p1.polynomial() == p2.polynomial()
True

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, default_prec=Integer(100), names=('x',)); (x,) = R._first_ngens(1)
>>> p = (Integer(1) + Integer(2)*x - x**Integer(4))**Integer(200)
>>> p1 = p.nth_root(Integer(1000), prec=Integer(100))
>>> p2 = (p.log()/Integer(1000)).exp()
>>> p1.prec() == p2.prec() == Integer(100)
True
>>> p1.polynomial() == p2.polynomial()
True

Sage Live

R.<x> = PowerSeriesRing(QQ, default_prec=100)
p = (1 + 2*x - x^4)**200
p1 = p.nth_root(1000, prec=100)
p2 = (p.log()/1000).exp()
p1.prec() == p2.prec() == 100
p1.polynomial() == p2.polynomial()

Positive characteristic:

Sage

sage: R.<u> = GF(3)[[]]
sage: p = 1 + 2 * u^2
sage: p.nth_root(4)
1 + 2*u^2 + u^6 + 2*u^8 + u^12 + 2*u^14 + O(u^20)
sage: p.nth_root(4)**4
1 + 2*u^2 + O(u^20)

Python

>>> from sage.all import *
>>> R = GF(Integer(3))[['u']]; (u,) = R._first_ngens(1)
>>> p = Integer(1) + Integer(2) * u**Integer(2)
>>> p.nth_root(Integer(4))
1 + 2*u^2 + u^6 + 2*u^8 + u^12 + 2*u^14 + O(u^20)
>>> p.nth_root(Integer(4))**Integer(4)
1 + 2*u^2 + O(u^20)

Sage Live

R.<u> = GF(3)[[]]
p = 1 + 2 * u^2
p.nth_root(4)
p.nth_root(4)**4

ogf_to_egf()[source]¶

Return the exponential generating function power series, assuming self is an ordinary generating function power series.

This is a formal Borel transform.

This can also be computed as serconvol(f,exp(t)) in PARI/GP.

See also

egf_to_ogf() for the inverse method.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ)
sage: f = t + t^2 + 2*t^3
sage: f.ogf_to_egf()
t + 1/2*t^2 + 1/3*t^3

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = t + t**Integer(2) + Integer(2)*t**Integer(3)
>>> f.ogf_to_egf()
t + 1/2*t^2 + 1/3*t^3

Sage Live

R.<t> = PowerSeriesRing(QQ)
f = t + t^2 + 2*t^3
f.ogf_to_egf()

padded_list(n=None)[source]¶

Return a list of coefficients of self up to (but not including) \(q^n\).

Includes 0s in the list on the right so that the list has length \(n\).

INPUT:

n – (optional) an integer that is at least 0. If n is not given, it will be taken to be the precision of self, unless this is +Infinity, in which case we just return self.list().

EXAMPLES:

Sage

sage: R.<q> = PowerSeriesRing(QQ)
sage: f = 1 - 17*q + 13*q^2 + 10*q^4 + O(q^7)
sage: f.list()
[1, -17, 13, 0, 10]
sage: f.padded_list(7)
[1, -17, 13, 0, 10, 0, 0]
sage: f.padded_list(10)
[1, -17, 13, 0, 10, 0, 0, 0, 0, 0]
sage: f.padded_list(3)
[1, -17, 13]
sage: f.padded_list()
[1, -17, 13, 0, 10, 0, 0]
sage: g = 1 - 17*q + 13*q^2 + 10*q^4
sage: g.list()
[1, -17, 13, 0, 10]
sage: g.padded_list()
[1, -17, 13, 0, 10]
sage: g.padded_list(10)
[1, -17, 13, 0, 10, 0, 0, 0, 0, 0]

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('q',)); (q,) = R._first_ngens(1)
>>> f = Integer(1) - Integer(17)*q + Integer(13)*q**Integer(2) + Integer(10)*q**Integer(4) + O(q**Integer(7))
>>> f.list()
[1, -17, 13, 0, 10]
>>> f.padded_list(Integer(7))
[1, -17, 13, 0, 10, 0, 0]
>>> f.padded_list(Integer(10))
[1, -17, 13, 0, 10, 0, 0, 0, 0, 0]
>>> f.padded_list(Integer(3))
[1, -17, 13]
>>> f.padded_list()
[1, -17, 13, 0, 10, 0, 0]
>>> g = Integer(1) - Integer(17)*q + Integer(13)*q**Integer(2) + Integer(10)*q**Integer(4)
>>> g.list()
[1, -17, 13, 0, 10]
>>> g.padded_list()
[1, -17, 13, 0, 10]
>>> g.padded_list(Integer(10))
[1, -17, 13, 0, 10, 0, 0, 0, 0, 0]

Sage Live

R.<q> = PowerSeriesRing(QQ)
f = 1 - 17*q + 13*q^2 + 10*q^4 + O(q^7)
f.list()
f.padded_list(7)
f.padded_list(10)
f.padded_list(3)
f.padded_list()
g = 1 - 17*q + 13*q^2 + 10*q^4
g.list()
g.padded_list()
g.padded_list(10)

polynomial()[source]¶

See this method in derived classes:

Implementations MUST override this in the derived class.

EXAMPLES:

Sage

sage: from sage.rings.power_series_ring_element import PowerSeries
sage: R.<x> = PowerSeriesRing(ZZ)
sage: PowerSeries.polynomial(1 + x^2)
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> from sage.rings.power_series_ring_element import PowerSeries
>>> R = PowerSeriesRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> PowerSeries.polynomial(Integer(1) + x**Integer(2))
Traceback (most recent call last):
...
NotImplementedError

Sage Live

from sage.rings.power_series_ring_element import PowerSeries
R.<x> = PowerSeriesRing(ZZ)
PowerSeries.polynomial(1 + x^2)

prec()[source]¶

The precision of \(...+O(x^r)\) is by definition \(r\).

EXAMPLES:

Sage

sage: R.<t> = ZZ[[]]
sage: (t^2 + O(t^3)).prec()
3
sage: (1 - t^2 + O(t^100)).prec()
100

Python

>>> from sage.all import *
>>> R = ZZ[['t']]; (t,) = R._first_ngens(1)
>>> (t**Integer(2) + O(t**Integer(3))).prec()
3
>>> (Integer(1) - t**Integer(2) + O(t**Integer(100))).prec()
100

Sage Live

R.<t> = ZZ[[]]
(t^2 + O(t^3)).prec()
(1 - t^2 + O(t^100)).prec()

precision_absolute()[source]¶

Return the absolute precision of this series.

By definition, the absolute precision of \(...+O(x^r)\) is \(r\).

EXAMPLES:

Sage

sage: R.<t> = ZZ[[]]
sage: (t^2 + O(t^3)).precision_absolute()
3
sage: (1 - t^2 + O(t^100)).precision_absolute()
100

Python

>>> from sage.all import *
>>> R = ZZ[['t']]; (t,) = R._first_ngens(1)
>>> (t**Integer(2) + O(t**Integer(3))).precision_absolute()
3
>>> (Integer(1) - t**Integer(2) + O(t**Integer(100))).precision_absolute()
100

Sage Live

R.<t> = ZZ[[]]
(t^2 + O(t^3)).precision_absolute()
(1 - t^2 + O(t^100)).precision_absolute()

precision_relative()[source]¶

Return the relative precision of this series, that is the difference between its absolute precision and its valuation.

By convention, the relative precision of \(0\) (or \(O(x^r)\) for any \(r\)) is \(0\).

EXAMPLES:

Sage

sage: R.<t> = ZZ[[]]
sage: (t^2 + O(t^3)).precision_relative()
1
sage: (1 - t^2 + O(t^100)).precision_relative()
100
sage: O(t^4).precision_relative()
0

Python

>>> from sage.all import *
>>> R = ZZ[['t']]; (t,) = R._first_ngens(1)
>>> (t**Integer(2) + O(t**Integer(3))).precision_relative()
1
>>> (Integer(1) - t**Integer(2) + O(t**Integer(100))).precision_relative()
100
>>> O(t**Integer(4)).precision_relative()
0

Sage Live

R.<t> = ZZ[[]]
(t^2 + O(t^3)).precision_relative()
(1 - t^2 + O(t^100)).precision_relative()
O(t^4).precision_relative()

shift(n)[source]¶

Return this power series multiplied by the power \(t^n\).

If \(n\) is negative, terms below \(t^{-n}\) are discarded.

This power series is left unchanged.

Note

Despite the fact that higher order terms are printed to the right in a power series, right shifting decreases the powers of \(t\), while left shifting increases them. This is to be consistent with polynomials, integers, etc.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ['y'], 't', 5)
sage: f = ~(1+t); f
1 - t + t^2 - t^3 + t^4 + O(t^5)
sage: f.shift(3)
t^3 - t^4 + t^5 - t^6 + t^7 + O(t^8)
sage: f >> 2
1 - t + t^2 + O(t^3)
sage: f << 10
t^10 - t^11 + t^12 - t^13 + t^14 + O(t^15)
sage: t << 29
t^30

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ['y'], 't', Integer(5), names=('t',)); (t,) = R._first_ngens(1)
>>> f = ~(Integer(1)+t); f
1 - t + t^2 - t^3 + t^4 + O(t^5)
>>> f.shift(Integer(3))
t^3 - t^4 + t^5 - t^6 + t^7 + O(t^8)
>>> f >> Integer(2)
1 - t + t^2 + O(t^3)
>>> f << Integer(10)
t^10 - t^11 + t^12 - t^13 + t^14 + O(t^15)
>>> t << Integer(29)
t^30

Sage Live

R.<t> = PowerSeriesRing(QQ['y'], 't', 5)
f = ~(1+t); f
f.shift(3)
f >> 2
f << 10
t << 29

AUTHORS:

Robert Bradshaw (2007-04-18)

sin(prec='infinity')[source]¶

Apply sin to the formal power series.

INPUT:

prec – integer or infinity; the degree to truncate the result to

OUTPUT: a new power series

EXAMPLES:

For one variable:

Sage

sage: t = PowerSeriesRing(QQ, 't').gen()
sage: f = (t + t**2).O(4)
sage: sin(f)
t + t^2 - 1/6*t^3 + O(t^4)

Python

>>> from sage.all import *
>>> t = PowerSeriesRing(QQ, 't').gen()
>>> f = (t + t**Integer(2)).O(Integer(4))
>>> sin(f)
t + t^2 - 1/6*t^3 + O(t^4)

Sage Live

t = PowerSeriesRing(QQ, 't').gen()
f = (t + t**2).O(4)
sin(f)

For several variables:

Sage

sage: T.<a,b> = PowerSeriesRing(ZZ,2)
sage: f = a + b + a*b + T.O(3)
sage: sin(f)
a + b + a*b + O(a, b)^3
sage: f.sin()
a + b + a*b + O(a, b)^3
sage: f.sin(prec=2)
a + b + O(a, b)^2

Python

>>> from sage.all import *
>>> T = PowerSeriesRing(ZZ,Integer(2), names=('a', 'b',)); (a, b,) = T._first_ngens(2)
>>> f = a + b + a*b + T.O(Integer(3))
>>> sin(f)
a + b + a*b + O(a, b)^3
>>> f.sin()
a + b + a*b + O(a, b)^3
>>> f.sin(prec=Integer(2))
a + b + O(a, b)^2

Sage Live

T.<a,b> = PowerSeriesRing(ZZ,2)
f = a + b + a*b + T.O(3)
sin(f)
f.sin()
f.sin(prec=2)

If the power series has a nonzero constant coefficient \(c\), one raises an error:

Sage

sage: g = 2+f
sage: sin(g)
Traceback (most recent call last):
...
ValueError: can only apply sin to formal power series with zero constant term

Python

>>> from sage.all import *
>>> g = Integer(2)+f
>>> sin(g)
Traceback (most recent call last):
...
ValueError: can only apply sin to formal power series with zero constant term

Sage Live

g = 2+f
sin(g)

If no precision is specified, the default precision is used:

Sage

sage: T.default_prec()
12
sage: sin(a)
a - 1/6*a^3 + 1/120*a^5 - 1/5040*a^7 + 1/362880*a^9 - 1/39916800*a^11 + O(a, b)^12
sage: a.sin(prec=5)
a - 1/6*a^3 + O(a, b)^5
sage: sin(a + T.O(5))
a - 1/6*a^3 + O(a, b)^5

Python

>>> from sage.all import *
>>> T.default_prec()
12
>>> sin(a)
a - 1/6*a^3 + 1/120*a^5 - 1/5040*a^7 + 1/362880*a^9 - 1/39916800*a^11 + O(a, b)^12
>>> a.sin(prec=Integer(5))
a - 1/6*a^3 + O(a, b)^5
>>> sin(a + T.O(Integer(5)))
a - 1/6*a^3 + O(a, b)^5

Sage Live

T.default_prec()
sin(a)
a.sin(prec=5)
sin(a + T.O(5))

sinh(prec='infinity')[source]¶

Apply sinh to the formal power series.

INPUT:

prec – integer or infinity; the degree to truncate the result to

OUTPUT: a new power series

EXAMPLES:

For one variable:

Sage

sage: t = PowerSeriesRing(QQ, 't').gen()
sage: f = (t + t**2).O(4)
sage: sinh(f)
t + t^2 + 1/6*t^3 + O(t^4)

Python

>>> from sage.all import *
>>> t = PowerSeriesRing(QQ, 't').gen()
>>> f = (t + t**Integer(2)).O(Integer(4))
>>> sinh(f)
t + t^2 + 1/6*t^3 + O(t^4)

Sage Live

t = PowerSeriesRing(QQ, 't').gen()
f = (t + t**2).O(4)
sinh(f)

For several variables:

Sage

sage: T.<a,b> = PowerSeriesRing(ZZ,2)
sage: f = a + b + a*b + T.O(3)
sage: sinh(f)
a + b + a*b + O(a, b)^3
sage: f.sinh()
a + b + a*b + O(a, b)^3
sage: f.sinh(prec=2)
a + b + O(a, b)^2

Python

>>> from sage.all import *
>>> T = PowerSeriesRing(ZZ,Integer(2), names=('a', 'b',)); (a, b,) = T._first_ngens(2)
>>> f = a + b + a*b + T.O(Integer(3))
>>> sinh(f)
a + b + a*b + O(a, b)^3
>>> f.sinh()
a + b + a*b + O(a, b)^3
>>> f.sinh(prec=Integer(2))
a + b + O(a, b)^2

Sage Live

T.<a,b> = PowerSeriesRing(ZZ,2)
f = a + b + a*b + T.O(3)
sinh(f)
f.sinh()
f.sinh(prec=2)

If the power series has a nonzero constant coefficient \(c\), one raises an error:

Sage

sage: g = 2 + f
sage: sinh(g)
Traceback (most recent call last):
...
ValueError: can only apply sinh to formal power series with zero
constant term

Python

>>> from sage.all import *
>>> g = Integer(2) + f
>>> sinh(g)
Traceback (most recent call last):
...
ValueError: can only apply sinh to formal power series with zero
constant term

Sage Live

g = 2 + f
sinh(g)

If no precision is specified, the default precision is used:

Sage

sage: T.default_prec()
12
sage: sinh(a)
a + 1/6*a^3 + 1/120*a^5 + 1/5040*a^7 + 1/362880*a^9 +
 1/39916800*a^11 + O(a, b)^12
sage: a.sinh(prec=5)
a + 1/6*a^3 + O(a, b)^5
sage: sinh(a + T.O(5))
a + 1/6*a^3 + O(a, b)^5

Python

>>> from sage.all import *
>>> T.default_prec()
12
>>> sinh(a)
a + 1/6*a^3 + 1/120*a^5 + 1/5040*a^7 + 1/362880*a^9 +
 1/39916800*a^11 + O(a, b)^12
>>> a.sinh(prec=Integer(5))
a + 1/6*a^3 + O(a, b)^5
>>> sinh(a + T.O(Integer(5)))
a + 1/6*a^3 + O(a, b)^5

Sage Live

T.default_prec()
sinh(a)
a.sinh(prec=5)
sinh(a + T.O(5))

solve_linear_de(prec='infinity', b=None, f0=None)[source]¶

Obtain a power series solution to an inhomogeneous linear differential equation of the form:

\[f'(t) = a(t) f(t) + b(t).\]

INPUT:

self – the power series \(a(t)\)
b – the power series \(b(t)\) (default: zero)
f0 – the constant term of \(f\) (“initial condition”) (default: 1)
prec – desired precision of result (this will be reduced if either a or b have less precision available)

OUTPUT: the power series \(f\), to indicated precision

ALGORITHM: A divide-and-conquer strategy; see the source code. Running time is approximately \(M(n) \log n\), where \(M(n)\) is the time required for a polynomial multiplication of length \(n\) over the coefficient ring. (If you’re working over something like \(\QQ\), running time analysis can be a little complicated because the coefficients tend to explode.)

Note

If the coefficient ring is a field of characteristic zero, then the solution will exist and is unique.
For other coefficient rings, things are more complicated. A solution may not exist, and if it does it may not be unique. Generally, by the time the \(n\)-th term has been computed, the algorithm will have attempted divisions by \(n!\) in the coefficient ring. So if your coefficient ring has enough ‘precision’, and if your coefficient ring can perform divisions even when the answer is not unique, and if you know in advance that a solution exists, then this function will find a solution (otherwise it will probably crash).

AUTHORS:

David Harvey (2006-09-11): factored functionality out from exp() function, cleaned up precision tests a bit

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ, default_prec=10)

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, default_prec=Integer(10), names=('t',)); (t,) = R._first_ngens(1)

Sage Live

R.<t> = PowerSeriesRing(QQ, default_prec=10)

Sage

sage: a = 2 - 3*t + 4*t^2 + O(t^10)
sage: b = 3 - 4*t^2 + O(t^7)
sage: f = a.solve_linear_de(prec=5, b=b, f0=3/5)
sage: f
 3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)
sage: f.derivative() - a*f - b
 O(t^4)

Python

>>> from sage.all import *
>>> a = Integer(2) - Integer(3)*t + Integer(4)*t**Integer(2) + O(t**Integer(10))
>>> b = Integer(3) - Integer(4)*t**Integer(2) + O(t**Integer(7))
>>> f = a.solve_linear_de(prec=Integer(5), b=b, f0=Integer(3)/Integer(5))
>>> f
 3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)
>>> f.derivative() - a*f - b
 O(t^4)

Sage Live

a = 2 - 3*t + 4*t^2 + O(t^10)
b = 3 - 4*t^2 + O(t^7)
f = a.solve_linear_de(prec=5, b=b, f0=3/5)
f
f.derivative() - a*f - b

Python

>>> from sage.all import *
>>> a = Integer(2) - Integer(3)*t + Integer(4)*t**Integer(2) + O(t**Integer(10))
>>> b = Integer(3) - Integer(4)*t**Integer(2) + O(t**Integer(7))
>>> f = a.solve_linear_de(prec=Integer(5), b=b, f0=Integer(3)/Integer(5))
>>> f
 3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)
>>> f.derivative() - a*f - b
 O(t^4)

Sage Live

a = 2 - 3*t + 4*t^2 + O(t^10)
b = 3 - 4*t^2 + O(t^7)
f = a.solve_linear_de(prec=5, b=b, f0=3/5)
f
f.derivative() - a*f - b

Sage

sage: a = 2 - 3*t + 4*t^2
sage: b = b = 3 - 4*t^2
sage: f = a.solve_linear_de(b=b, f0=3/5)
Traceback (most recent call last):
...
ValueError: cannot solve differential equation to infinite precision

Python

>>> from sage.all import *
>>> a = Integer(2) - Integer(3)*t + Integer(4)*t**Integer(2)
>>> b = b = Integer(3) - Integer(4)*t**Integer(2)
>>> f = a.solve_linear_de(b=b, f0=Integer(3)/Integer(5))
Traceback (most recent call last):
...
ValueError: cannot solve differential equation to infinite precision

Sage Live

a = 2 - 3*t + 4*t^2
b = b = 3 - 4*t^2
f = a.solve_linear_de(b=b, f0=3/5)

Python

>>> from sage.all import *
>>> a = Integer(2) - Integer(3)*t + Integer(4)*t**Integer(2)
>>> b = b = Integer(3) - Integer(4)*t**Integer(2)
>>> f = a.solve_linear_de(b=b, f0=Integer(3)/Integer(5))
Traceback (most recent call last):
...
ValueError: cannot solve differential equation to infinite precision

Sage Live

a = 2 - 3*t + 4*t^2
b = b = 3 - 4*t^2
f = a.solve_linear_de(b=b, f0=3/5)

Sage

sage: a.solve_linear_de(prec=5, b=b, f0=3/5)
 3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)

Python

>>> from sage.all import *
>>> a.solve_linear_de(prec=Integer(5), b=b, f0=Integer(3)/Integer(5))
 3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)

Sage Live

a.solve_linear_de(prec=5, b=b, f0=3/5)

Python

>>> from sage.all import *
>>> a.solve_linear_de(prec=Integer(5), b=b, f0=Integer(3)/Integer(5))
 3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)

Sage Live

a.solve_linear_de(prec=5, b=b, f0=3/5)

sqrt(prec=None, extend=False, all=False, name=None)[source]¶

Return a square root of self.

INPUT:

prec – integer (default: None); if not None and the series has infinite precision, truncates series at precision prec

extend – boolean (default: False); if True, return a square root in an extension ring, if necessary. Otherwise, raise a ValueError if the square root is not in the base power series ring. For example, if extend is True, the square root of a power series with odd degree leading coefficient is defined as an element of a formal extension ring.

name – string; if extend is True, you must also specify the print name of the formal square root

all – boolean (default: False); if True, return all square roots of self, instead of just one

ALGORITHM: Newton’s method

\[x_{i+1} = \frac{1}{2}( x_i + \mathrm{self}/x_i )\]

EXAMPLES:

Sage

sage: K.<t> = PowerSeriesRing(QQ, 't', 5)
sage: sqrt(t^2)
t
sage: sqrt(1 + t)
1 + 1/2*t - 1/8*t^2 + 1/16*t^3 - 5/128*t^4 + O(t^5)
sage: sqrt(4 + t)
2 + 1/4*t - 1/64*t^2 + 1/512*t^3 - 5/16384*t^4 + O(t^5)
sage: u = sqrt(2 + t, prec=2, extend=True, name = 'alpha'); u
alpha
sage: u^2
2 + t
sage: u.parent()
Univariate Quotient Polynomial Ring in alpha
 over Power Series Ring in t over Rational Field
 with modulus x^2 - 2 - t
sage: K.<t> = PowerSeriesRing(QQ, 't', 50)
sage: sqrt(1 + 2*t + t^2)
1 + t
sage: sqrt(t^2 + 2*t^4 + t^6)
t + t^3
sage: sqrt(1 + t + t^2 + 7*t^3)^2
1 + t + t^2 + 7*t^3 + O(t^50)
sage: sqrt(K(0))
0
sage: sqrt(t^2)
t

Python

>>> from sage.all import *
>>> K = PowerSeriesRing(QQ, 't', Integer(5), names=('t',)); (t,) = K._first_ngens(1)
>>> sqrt(t**Integer(2))
t
>>> sqrt(Integer(1) + t)
1 + 1/2*t - 1/8*t^2 + 1/16*t^3 - 5/128*t^4 + O(t^5)
>>> sqrt(Integer(4) + t)
2 + 1/4*t - 1/64*t^2 + 1/512*t^3 - 5/16384*t^4 + O(t^5)
>>> u = sqrt(Integer(2) + t, prec=Integer(2), extend=True, name = 'alpha'); u
alpha
>>> u**Integer(2)
2 + t
>>> u.parent()
Univariate Quotient Polynomial Ring in alpha
 over Power Series Ring in t over Rational Field
 with modulus x^2 - 2 - t
>>> K = PowerSeriesRing(QQ, 't', Integer(50), names=('t',)); (t,) = K._first_ngens(1)
>>> sqrt(Integer(1) + Integer(2)*t + t**Integer(2))
1 + t
>>> sqrt(t**Integer(2) + Integer(2)*t**Integer(4) + t**Integer(6))
t + t^3
>>> sqrt(Integer(1) + t + t**Integer(2) + Integer(7)*t**Integer(3))**Integer(2)
1 + t + t^2 + 7*t^3 + O(t^50)
>>> sqrt(K(Integer(0)))
0
>>> sqrt(t**Integer(2))
t

Sage Live

K.<t> = PowerSeriesRing(QQ, 't', 5)
sqrt(t^2)
sqrt(1 + t)
sqrt(4 + t)
u = sqrt(2 + t, prec=2, extend=True, name = 'alpha'); u
u^2
u.parent()
K.<t> = PowerSeriesRing(QQ, 't', 50)
sqrt(1 + 2*t + t^2)
sqrt(t^2 + 2*t^4 + t^6)
sqrt(1 + t + t^2 + 7*t^3)^2
sqrt(K(0))
sqrt(t^2)

Sage

sage: # needs sage.rings.complex_double
sage: K.<t> = PowerSeriesRing(CDF, 5)
sage: v = sqrt(-1 + t + t^3, all=True); v
[1.0*I - 0.5*I*t - 0.125*I*t^2 - 0.5625*I*t^3 - 0.2890625*I*t^4 + O(t^5),
 -1.0*I + 0.5*I*t + 0.125*I*t^2 + 0.5625*I*t^3 + 0.2890625*I*t^4 + O(t^5)]
sage: [a^2 for a in v]
[-1.0 + 1.0*t + 0.0*t^2 + 1.0*t^3 + O(t^5), -1.0 + 1.0*t + 0.0*t^2 + 1.0*t^3 + O(t^5)]

Python

>>> from sage.all import *
>>> # needs sage.rings.complex_double
>>> K = PowerSeriesRing(CDF, Integer(5), names=('t',)); (t,) = K._first_ngens(1)
>>> v = sqrt(-Integer(1) + t + t**Integer(3), all=True); v
[1.0*I - 0.5*I*t - 0.125*I*t^2 - 0.5625*I*t^3 - 0.2890625*I*t^4 + O(t^5),
 -1.0*I + 0.5*I*t + 0.125*I*t^2 + 0.5625*I*t^3 + 0.2890625*I*t^4 + O(t^5)]
>>> [a**Integer(2) for a in v]
[-1.0 + 1.0*t + 0.0*t^2 + 1.0*t^3 + O(t^5), -1.0 + 1.0*t + 0.0*t^2 + 1.0*t^3 + O(t^5)]

Sage Live

# needs sage.rings.complex_double
K.<t> = PowerSeriesRing(CDF, 5)
v = sqrt(-1 + t + t^3, all=True); v
[a^2 for a in v]

Python

>>> from sage.all import *
>>> # needs sage.rings.complex_double
>>> K = PowerSeriesRing(CDF, Integer(5), names=('t',)); (t,) = K._first_ngens(1)
>>> v = sqrt(-Integer(1) + t + t**Integer(3), all=True); v
[1.0*I - 0.5*I*t - 0.125*I*t^2 - 0.5625*I*t^3 - 0.2890625*I*t^4 + O(t^5),
 -1.0*I + 0.5*I*t + 0.125*I*t^2 + 0.5625*I*t^3 + 0.2890625*I*t^4 + O(t^5)]
>>> [a**Integer(2) for a in v]
[-1.0 + 1.0*t + 0.0*t^2 + 1.0*t^3 + O(t^5), -1.0 + 1.0*t + 0.0*t^2 + 1.0*t^3 + O(t^5)]

Sage Live

# needs sage.rings.complex_double
K.<t> = PowerSeriesRing(CDF, 5)
v = sqrt(-1 + t + t^3, all=True); v
[a^2 for a in v]

A formal square root:

Sage

sage: K.<t> = PowerSeriesRing(QQ, 5)
sage: f = 2*t + t^3 + O(t^4)
sage: s = f.sqrt(extend=True, name='sqrtf'); s
sqrtf
sage: s^2
2*t + t^3 + O(t^4)
sage: parent(s)
Univariate Quotient Polynomial Ring in sqrtf
 over Power Series Ring in t over Rational Field
 with modulus x^2 - 2*t - t^3 + O(t^4)

Python

>>> from sage.all import *
>>> K = PowerSeriesRing(QQ, Integer(5), names=('t',)); (t,) = K._first_ngens(1)
>>> f = Integer(2)*t + t**Integer(3) + O(t**Integer(4))
>>> s = f.sqrt(extend=True, name='sqrtf'); s
sqrtf
>>> s**Integer(2)
2*t + t^3 + O(t^4)
>>> parent(s)
Univariate Quotient Polynomial Ring in sqrtf
 over Power Series Ring in t over Rational Field
 with modulus x^2 - 2*t - t^3 + O(t^4)

Sage Live

K.<t> = PowerSeriesRing(QQ, 5)
f = 2*t + t^3 + O(t^4)
s = f.sqrt(extend=True, name='sqrtf'); s
s^2
parent(s)

AUTHORS:

Robert Bradshaw
William Stein

square_root()[source]¶

Return the square root of self in this ring. If this cannot be done, then an error will be raised.

This function succeeds if and only if self. is_square()

EXAMPLES:

Sage

sage: K.<t> = PowerSeriesRing(QQ, 't', 5)
sage: (1 + t).square_root()
1 + 1/2*t - 1/8*t^2 + 1/16*t^3 - 5/128*t^4 + O(t^5)
sage: (2 + t).square_root()
Traceback (most recent call last):
...
ValueError: Square root does not live in this ring.
sage: (2 + t.change_ring(RR)).square_root()                                 # needs sage.rings.real_mpfr
1.41421356237309 + 0.353553390593274*t - 0.0441941738241592*t^2
 + 0.0110485434560398*t^3 - 0.00345266983001244*t^4 + O(t^5)
sage: t.square_root()
Traceback (most recent call last):
...
ValueError: Square root not defined for power series of odd valuation.
sage: K.<t> = PowerSeriesRing(ZZ, 't', 5)
sage: f = (1+t)^20
sage: f.square_root()
1 + 10*t + 45*t^2 + 120*t^3 + 210*t^4 + O(t^5)
sage: f = 1 + t
sage: f.square_root()
Traceback (most recent call last):
...
ValueError: Square root does not live in this ring.

Python

>>> from sage.all import *
>>> K = PowerSeriesRing(QQ, 't', Integer(5), names=('t',)); (t,) = K._first_ngens(1)
>>> (Integer(1) + t).square_root()
1 + 1/2*t - 1/8*t^2 + 1/16*t^3 - 5/128*t^4 + O(t^5)
>>> (Integer(2) + t).square_root()
Traceback (most recent call last):
...
ValueError: Square root does not live in this ring.
>>> (Integer(2) + t.change_ring(RR)).square_root()                                 # needs sage.rings.real_mpfr
1.41421356237309 + 0.353553390593274*t - 0.0441941738241592*t^2
 + 0.0110485434560398*t^3 - 0.00345266983001244*t^4 + O(t^5)
>>> t.square_root()
Traceback (most recent call last):
...
ValueError: Square root not defined for power series of odd valuation.
>>> K = PowerSeriesRing(ZZ, 't', Integer(5), names=('t',)); (t,) = K._first_ngens(1)
>>> f = (Integer(1)+t)**Integer(20)
>>> f.square_root()
1 + 10*t + 45*t^2 + 120*t^3 + 210*t^4 + O(t^5)
>>> f = Integer(1) + t
>>> f.square_root()
Traceback (most recent call last):
...
ValueError: Square root does not live in this ring.

Sage Live

K.<t> = PowerSeriesRing(QQ, 't', 5)
(1 + t).square_root()
(2 + t).square_root()
(2 + t.change_ring(RR)).square_root()                                 # needs sage.rings.real_mpfr
t.square_root()
K.<t> = PowerSeriesRing(ZZ, 't', 5)
f = (1+t)^20
f.square_root()
f = 1 + t
f.square_root()

AUTHORS:

Robert Bradshaw

stieltjes_continued_fraction()[source]¶

Return the Stieltjes continued fraction of self.

The S-fraction or Stieltjes continued fraction of a power series is a continued fraction expansion with steps of size one. We use the following convention

\[1 / (1 - A_1 t / (1 - A_2 t / (1 - A_3 t / (1 - \cdots))))\]

OUTPUT: \(A_n\) for \(n \geq 1\)

The expansion is done as long as possible given the precision. Whenever the expansion is not well-defined, because it would require to divide by zero, an exception is raised.

EXAMPLES:

Sage

sage: t = PowerSeriesRing(QQ, 't').gen()
sage: s = sum(catalan_number(k) * t**k for k in range(12)).O(12)
sage: s.stieltjes_continued_fraction()
(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

Python

>>> from sage.all import *
>>> t = PowerSeriesRing(QQ, 't').gen()
>>> s = sum(catalan_number(k) * t**k for k in range(Integer(12))).O(Integer(12))
>>> s.stieltjes_continued_fraction()
(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

Sage Live

t = PowerSeriesRing(QQ, 't').gen()
s = sum(catalan_number(k) * t**k for k in range(12)).O(12)
s.stieltjes_continued_fraction()

Another example:

Sage

sage: (exp(t)).stieltjes_continued_fraction()
(1,
 -1/2,
 1/6,
 -1/6,
 1/10,
 -1/10,
 1/14,
 -1/14,
 1/18,
 -1/18,
 1/22,
 -1/22,
 1/26,
 -1/26,
 1/30,
 -1/30,
 1/34,
 -1/34,
 1/38)

Python

>>> from sage.all import *
>>> (exp(t)).stieltjes_continued_fraction()
(1,
 -1/2,
 1/6,
 -1/6,
 1/10,
 -1/10,
 1/14,
 -1/14,
 1/18,
 -1/18,
 1/22,
 -1/22,
 1/26,
 -1/26,
 1/30,
 -1/30,
 1/34,
 -1/34,
 1/38)

Sage Live

(exp(t)).stieltjes_continued_fraction()

tan(prec='infinity')[source]¶

Apply tan to the formal power series.

INPUT:

prec – integer or infinity; the degree to truncate the result to

OUTPUT: a new power series

EXAMPLES:

For one variable:

Sage

sage: t = PowerSeriesRing(QQ, 't').gen()
sage: f = (t + t**2).O(4)
sage: tan(f)
t + t^2 + 1/3*t^3 + O(t^4)

Python

>>> from sage.all import *
>>> t = PowerSeriesRing(QQ, 't').gen()
>>> f = (t + t**Integer(2)).O(Integer(4))
>>> tan(f)
t + t^2 + 1/3*t^3 + O(t^4)

Sage Live

t = PowerSeriesRing(QQ, 't').gen()
f = (t + t**2).O(4)
tan(f)

For several variables:

Sage

sage: T.<a,b> = PowerSeriesRing(ZZ,2)
sage: f = a + b + a*b + T.O(3)
sage: tan(f)
a + b + a*b + O(a, b)^3
sage: f.tan()
a + b + a*b + O(a, b)^3
sage: f.tan(prec=2)
a + b + O(a, b)^2

Python

>>> from sage.all import *
>>> T = PowerSeriesRing(ZZ,Integer(2), names=('a', 'b',)); (a, b,) = T._first_ngens(2)
>>> f = a + b + a*b + T.O(Integer(3))
>>> tan(f)
a + b + a*b + O(a, b)^3
>>> f.tan()
a + b + a*b + O(a, b)^3
>>> f.tan(prec=Integer(2))
a + b + O(a, b)^2

Sage Live

T.<a,b> = PowerSeriesRing(ZZ,2)
f = a + b + a*b + T.O(3)
tan(f)
f.tan()
f.tan(prec=2)

If the power series has a nonzero constant coefficient \(c\), one raises an error:

Sage

sage: g = 2 + f
sage: tan(g)
Traceback (most recent call last):
...
ValueError: can only apply tan to formal power series with zero constant term

Python

>>> from sage.all import *
>>> g = Integer(2) + f
>>> tan(g)
Traceback (most recent call last):
...
ValueError: can only apply tan to formal power series with zero constant term

Sage Live

g = 2 + f
tan(g)

If no precision is specified, the default precision is used:

Sage

sage: T.default_prec()
12
sage: tan(a)
a + 1/3*a^3 + 2/15*a^5 + 17/315*a^7 + 62/2835*a^9 + 1382/155925*a^11 + O(a, b)^12
sage: a.tan(prec=5)
a + 1/3*a^3 + O(a, b)^5
sage: tan(a + T.O(5))
a + 1/3*a^3 + O(a, b)^5

Python

>>> from sage.all import *
>>> T.default_prec()
12
>>> tan(a)
a + 1/3*a^3 + 2/15*a^5 + 17/315*a^7 + 62/2835*a^9 + 1382/155925*a^11 + O(a, b)^12
>>> a.tan(prec=Integer(5))
a + 1/3*a^3 + O(a, b)^5
>>> tan(a + T.O(Integer(5)))
a + 1/3*a^3 + O(a, b)^5

Sage Live

T.default_prec()
tan(a)
a.tan(prec=5)
tan(a + T.O(5))

tanh(prec='infinity')[source]¶

Apply tanh to the formal power series.

INPUT:

prec – integer or infinity; the degree to truncate the result to

OUTPUT: a new power series

EXAMPLES:

For one variable:

Sage

sage: t = PowerSeriesRing(QQ, 't').gen()
sage: f = (t + t**2).O(4)
sage: tanh(f)
t + t^2 - 1/3*t^3 + O(t^4)

Python

>>> from sage.all import *
>>> t = PowerSeriesRing(QQ, 't').gen()
>>> f = (t + t**Integer(2)).O(Integer(4))
>>> tanh(f)
t + t^2 - 1/3*t^3 + O(t^4)

Sage Live

t = PowerSeriesRing(QQ, 't').gen()
f = (t + t**2).O(4)
tanh(f)

For several variables:

Sage

sage: T.<a,b> = PowerSeriesRing(ZZ,2)
sage: f = a + b + a*b + T.O(3)
sage: tanh(f)
a + b + a*b + O(a, b)^3
sage: f.tanh()
a + b + a*b + O(a, b)^3
sage: f.tanh(prec=2)
a + b + O(a, b)^2

Python

>>> from sage.all import *
>>> T = PowerSeriesRing(ZZ,Integer(2), names=('a', 'b',)); (a, b,) = T._first_ngens(2)
>>> f = a + b + a*b + T.O(Integer(3))
>>> tanh(f)
a + b + a*b + O(a, b)^3
>>> f.tanh()
a + b + a*b + O(a, b)^3
>>> f.tanh(prec=Integer(2))
a + b + O(a, b)^2

Sage Live

T.<a,b> = PowerSeriesRing(ZZ,2)
f = a + b + a*b + T.O(3)
tanh(f)
f.tanh()
f.tanh(prec=2)

If the power series has a nonzero constant coefficient \(c\), one raises an error:

Sage

sage: g = 2 + f
sage: tanh(g)
Traceback (most recent call last):
...
ValueError: can only apply tanh to formal power series with zero
 constant term

Python

>>> from sage.all import *
>>> g = Integer(2) + f
>>> tanh(g)
Traceback (most recent call last):
...
ValueError: can only apply tanh to formal power series with zero
 constant term

Sage Live

g = 2 + f
tanh(g)

If no precision is specified, the default precision is used:

Sage

sage: T.default_prec()
12
sage: tanh(a)
a - 1/3*a^3 + 2/15*a^5 - 17/315*a^7 + 62/2835*a^9 -
1382/155925*a^11 + O(a, b)^12
sage: a.tanh(prec=5)
a - 1/3*a^3 + O(a, b)^5
sage: tanh(a + T.O(5))
a - 1/3*a^3 + O(a, b)^5

Python

>>> from sage.all import *
>>> T.default_prec()
12
>>> tanh(a)
a - 1/3*a^3 + 2/15*a^5 - 17/315*a^7 + 62/2835*a^9 -
1382/155925*a^11 + O(a, b)^12
>>> a.tanh(prec=Integer(5))
a - 1/3*a^3 + O(a, b)^5
>>> tanh(a + T.O(Integer(5)))
a - 1/3*a^3 + O(a, b)^5

Sage Live

T.default_prec()
tanh(a)
a.tanh(prec=5)
tanh(a + T.O(5))

truncate(prec='infinity')[source]¶

The polynomial obtained from power series by truncation.

EXAMPLES:

Sage

sage: R.<I> = GF(2)[[]]
sage: f = 1/(1+I+O(I^8)); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
sage: f.truncate(5)
I^4 + I^3 + I^2 + I + 1

Python

>>> from sage.all import *
>>> R = GF(Integer(2))[['I']]; (I,) = R._first_ngens(1)
>>> f = Integer(1)/(Integer(1)+I+O(I**Integer(8))); f
1 + I + I^2 + I^3 + I^4 + I^5 + I^6 + I^7 + O(I^8)
>>> f.truncate(Integer(5))
I^4 + I^3 + I^2 + I + 1

Sage Live

R.<I> = GF(2)[[]]
f = 1/(1+I+O(I^8)); f
f.truncate(5)

valuation()[source]¶

Return the valuation of this power series.

This is equal to the valuation of the underlying polynomial.

EXAMPLES:

Sparse examples:

Sage

sage: R.<t> = PowerSeriesRing(QQ, sparse=True)
sage: f = t^100000 + O(t^10000000)
sage: f.valuation()
100000
sage: R(0).valuation()
+Infinity

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, sparse=True, names=('t',)); (t,) = R._first_ngens(1)
>>> f = t**Integer(100000) + O(t**Integer(10000000))
>>> f.valuation()
100000
>>> R(Integer(0)).valuation()
+Infinity

Sage Live

R.<t> = PowerSeriesRing(QQ, sparse=True)
f = t^100000 + O(t^10000000)
f.valuation()
R(0).valuation()

Dense examples:

Sage

sage: R.<t> = PowerSeriesRing(ZZ)
sage: f = 17*t^100 + O(t^110)
sage: f.valuation()
100
sage: t.valuation()
1

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('t',)); (t,) = R._first_ngens(1)
>>> f = Integer(17)*t**Integer(100) + O(t**Integer(110))
>>> f.valuation()
100
>>> t.valuation()
1

Sage Live

R.<t> = PowerSeriesRing(ZZ)
f = 17*t^100 + O(t^110)
f.valuation()
t.valuation()

valuation_zero_part()[source]¶

Factor self as \(q^n \cdot (a_0 + a_1 q + \cdots)\) with \(a_0\) nonzero. Then this function returns \(a_0 + a_1 q + \cdots\) .

Note

This valuation zero part need not be a unit if, e.g., \(a_0\) is not invertible in the base ring.

EXAMPLES:

Sage

sage: R.<t> = PowerSeriesRing(QQ)
sage: ((1/3)*t^5*(17-2/3*t^3)).valuation_zero_part()
17/3 - 2/9*t^3

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(QQ, names=('t',)); (t,) = R._first_ngens(1)
>>> ((Integer(1)/Integer(3))*t**Integer(5)*(Integer(17)-Integer(2)/Integer(3)*t**Integer(3))).valuation_zero_part()
17/3 - 2/9*t^3

Sage Live

R.<t> = PowerSeriesRing(QQ)
((1/3)*t^5*(17-2/3*t^3)).valuation_zero_part()

In this example the valuation 0 part is not a unit:

Sage

sage: R.<t> = PowerSeriesRing(ZZ, sparse=True)
sage: u = (-2*t^5*(17-t^3)).valuation_zero_part(); u
-34 + 2*t^3
sage: u.is_unit()
False
sage: u.valuation()
0

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, sparse=True, names=('t',)); (t,) = R._first_ngens(1)
>>> u = (-Integer(2)*t**Integer(5)*(Integer(17)-t**Integer(3))).valuation_zero_part(); u
-34 + 2*t^3
>>> u.is_unit()
False
>>> u.valuation()
0

Sage Live

R.<t> = PowerSeriesRing(ZZ, sparse=True)
u = (-2*t^5*(17-t^3)).valuation_zero_part(); u
u.is_unit()
u.valuation()

variable()[source]¶

Return a string with the name of the variable of this power series.

EXAMPLES:

Sage

sage: R.<x> = PowerSeriesRing(Rationals())
sage: f = x^2 + 3*x^4 + O(x^7)
sage: f.variable()
'x'

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(Rationals(), names=('x',)); (x,) = R._first_ngens(1)
>>> f = x**Integer(2) + Integer(3)*x**Integer(4) + O(x**Integer(7))
>>> f.variable()
'x'

Sage Live

R.<x> = PowerSeriesRing(Rationals())
f = x^2 + 3*x^4 + O(x^7)
f.variable()

AUTHORS:

David Harvey (2006-08-08)

sage.rings.power_series_ring_element.is_PowerSeries(x)[source]¶

Return True if x is an instance of a univariate or multivariate power series.

EXAMPLES:

Sage

sage: R.<x> = PowerSeriesRing(ZZ)
sage: from sage.rings.power_series_ring_element import is_PowerSeries
sage: is_PowerSeries(1 + x^2)
doctest:warning...
DeprecationWarning: The function is_PowerSeries is deprecated; use 'isinstance(..., PowerSeries)' instead.
See https://github.com/sagemath/sage/issues/38266 for details.
True
sage: is_PowerSeries(x - x)
True
sage: is_PowerSeries(0)
False
sage: var('x')                                                                  # needs sage.symbolic
x
sage: is_PowerSeries(1 + x^2)                                                   # needs sage.symbolic
False

Python

>>> from sage.all import *
>>> R = PowerSeriesRing(ZZ, names=('x',)); (x,) = R._first_ngens(1)
>>> from sage.rings.power_series_ring_element import is_PowerSeries
>>> is_PowerSeries(Integer(1) + x**Integer(2))
doctest:warning...
DeprecationWarning: The function is_PowerSeries is deprecated; use 'isinstance(..., PowerSeries)' instead.
See https://github.com/sagemath/sage/issues/38266 for details.
True
>>> is_PowerSeries(x - x)
True
>>> is_PowerSeries(Integer(0))
False
>>> var('x')                                                                  # needs sage.symbolic
x
>>> is_PowerSeries(Integer(1) + x**Integer(2))                                                   # needs sage.symbolic
False

Sage Live

R.<x> = PowerSeriesRing(ZZ)
from sage.rings.power_series_ring_element import is_PowerSeries
is_PowerSeries(1 + x^2)
is_PowerSeries(x - x)
is_PowerSeries(0)
var('x')                                                                  # needs sage.symbolic
is_PowerSeries(1 + x^2)                                                   # needs sage.symbolic

sage.rings.power_series_ring_element.make_element_from_parent_v0(parent, *args)[source]¶

sage.rings.power_series_ring_element.make_powerseries_poly_v0(parent, f, prec, is_gen)[source]¶