Lazy Series

A lazy series is a series whose coefficients are computed on demand. Therefore, unlike the usual Laurent/power/etc. series in Sage, lazy series have infinite precision.

EXAMPLES:

Laurent series over the integer ring are particularly useful as generating functions for sequences arising in combinatorics.

sage: L.<z> = LazyLaurentSeriesRing(ZZ)

The generating function of the Fibonacci sequence is:

sage: f = 1 / (1 - z - z^2)
sage: f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)

In principle, we can now compute any coefficient of \(f\):

sage: f.coefficient(100)
573147844013817084101

Which coefficients are actually computed depends on the type of implementation. For the sparse implementation, only the coefficients which are needed are computed.

sage: s = L(lambda n: n, valuation=0); s
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: s.coefficient(10)
10
sage: s._coeff_stream._cache
{0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 10: 10}

Using the dense implementation, all coefficients up to the required coefficient are computed.

sage: L.<x> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: s = L(lambda n: n, valuation=0); s
x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + O(x^7)
sage: s.coefficient(10)
10
sage: s._coeff_stream._cache
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

We can do arithmetic with lazy power series:

sage: f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
sage: f^-1
1 - z - z^2 + O(z^7)
sage: f + f^-1
2 + z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
sage: g = (f + f^-1)*(f - f^-1); g
4*z + 6*z^2 + 8*z^3 + 19*z^4 + 38*z^5 + 71*z^6 + O(z^7)

We can change the base ring:

sage: h = g.change_ring(QQ)
sage: h.parent()
Lazy Laurent Series Ring in z over Rational Field
sage: h
4*z + 6*z^2 + 8*z^3 + 19*z^4 + 38*z^5 + 71*z^6 + O(z^7)
sage: hinv = h^-1; hinv
1/4*z^-1 - 3/8 + 1/16*z - 17/32*z^2 + 5/64*z^3 - 29/128*z^4 + 165/256*z^5 + O(z^6)
sage: hinv.valuation()
-1

AUTHORS:

• Kwankyu Lee (2019-02-24): initial version

• Tejasvi Chebrolu, Martin Rubey, Travis Scrimshaw (2021-08): refactored and expanded functionality

class sage.rings.lazy_series.LazyCauchyProductSeries(parent, coeff_stream)

Bases: sage.rings.lazy_series.LazyModuleElement

A class for series where multiplication is the Cauchy product.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = 1 / (1 - z)
sage: f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: f * (1 - z)
1 + O(z^7)

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: f = 1 / (1 - z)
sage: f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)

valuation()

Return the valuation of self.

This method determines the valuation of the series by looking for a nonzero coefficient. Hence if the series happens to be zero, then it may run forever.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: s = 1/(1 - z) - 1/(1 - 2*z)
sage: s.valuation()
1
sage: t = z - z
sage: t.valuation()
+Infinity
sage: M = L(lambda n: n^2, 0)
sage: M.valuation()
1
sage: (M - M).valuation()
+Infinity

class sage.rings.lazy_series.LazyDirichletSeries(parent, coeff_stream)

Bases: sage.rings.lazy_series.LazyModuleElement

A Dirichlet series where the coefficients are computed lazily.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: f = L(constant=1)^2; f
1 + 2/2^z + 2/3^z + 3/4^z + 2/5^z + 4/6^z + 2/7^z + O(1/(8^z))
sage: f.coefficient(100) == number_of_divisors(100)
True

Lazy Dirichlet series is picklable:

sage: g = loads(dumps(f))
sage: g
1 + 2/2^z + 2/3^z + 3/4^z + 2/5^z + 4/6^z + 2/7^z + O(1/(8^z))
sage: g == f
True

valuation()

Return the valuation of self.

This method determines the valuation of the series by looking for a nonzero coefficient. Hence if the series happens to be zero, then it may run forever.

EXAMPLES:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: mu = L(moebius); mu.valuation()
0
sage: (mu - mu).valuation()
+Infinity
sage: g = L(constant=1, valuation=2)
sage: g.valuation()
log(2)
sage: (g*g).valuation()
2*log(2)

class sage.rings.lazy_series.LazyLaurentSeries(parent, coeff_stream)

Bases: sage.rings.lazy_series.LazyCauchyProductSeries

A Laurent series where the coefficients are computed lazily.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)

We can build a series from a function and specify if the series eventually takes a constant value:

sage: f = L(lambda i: i, valuation=-3, constant=-1, degree=3)
sage: f
-3*z^-3 - 2*z^-2 - z^-1 + z + 2*z^2 - z^3 - z^4 - z^5 + O(z^6)
sage: f[-2]
-2
sage: f[10]
-1
sage: f[-5]
0

sage: f = L(lambda i: i, valuation=-3)
sage: f
-3*z^-3 - 2*z^-2 - z^-1 + z + 2*z^2 + 3*z^3 + O(z^4)
sage: f[20]
20

Anything that converts into a polynomial can be input, where we can also specify the valuation or if the series eventually takes a constant value:

sage: L([-5,2,0,5])
-5 + 2*z + 5*z^3
sage: L([-5,2,0,5], constant=6)
-5 + 2*z + 5*z^3 + 6*z^4 + 6*z^5 + 6*z^6 + O(z^7)
sage: L([-5,2,0,5], degree=6, constant=6)
-5 + 2*z + 5*z^3 + 6*z^6 + 6*z^7 + 6*z^8 + O(z^9)
sage: L([-5,2,0,5], valuation=-2, degree=3, constant=6)
-5*z^-2 + 2*z^-1 + 5*z + 6*z^3 + 6*z^4 + 6*z^5 + O(z^6)
sage: L([-5,2,0,5], valuation=5)
-5*z^5 + 2*z^6 + 5*z^8
sage: L({-2:9, 3:4}, constant=2, degree=5)
9*z^-2 + 4*z^3 + 2*z^5 + 2*z^6 + 2*z^7 + O(z^8)

We can also perform arithmetic:

sage: f = 1 / (1 - z - z^2)
sage: f
1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + O(z^7)
sage: f.coefficient(100)
573147844013817084101
sage: f = (z^-2 - 1 + 2*z) / (z^-1 - z + 3*z^2)
sage: f
z^-1 - z^2 - z^4 + 3*z^5 + O(z^6)

However, we may not always be able to know when a result is exactly a polynomial:

sage: f * (z^-1 - z + 3*z^2)
z^-2 - 1 + 2*z + O(z^5)

approximate_series(prec, name=None)

Return the Laurent series with absolute precision prec approximated from this series.

INPUT:

• prec – an integer

• name – name of the variable; if it is None, the name of the variable of the series is used

OUTPUT: a Laurent series with absolute precision prec

EXAMPLES:

sage: L = LazyLaurentSeriesRing(ZZ, 'z')
sage: z = L.gen()
sage: f = (z - 2*z^3)^5/(1 - 2*z)
sage: f
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + 32*z^10 - 16*z^11 + O(z^12)
sage: g = f.approximate_series(10)
sage: g
z^5 + 2*z^6 - 6*z^7 - 12*z^8 + 16*z^9 + O(z^10)
sage: g.parent()
Power Series Ring in z over Integer Ring
sage: h = (f^-1).approximate_series(3)
sage: h
z^-5 - 2*z^-4 + 10*z^-3 - 20*z^-2 + 60*z^-1 - 120 + 280*z - 560*z^2 + O(z^3)
sage: h.parent()
Laurent Series Ring in z over Integer Ring

compose(g)

Return the composition of self with g.

Given two Laurent Series \(f\) and \(g\) over the same base ring, the composition \((f \circ g)(z) = f(g(z))\) is defined if and only if:

• \(g = 0\) and \(val(f) >= 0\),

• \(g\) is non-zero and \(f\) has only finitely many non-zero coefficients,

• \(g\) is non-zero and \(val(g) > 0\).

INPUT:

• g – other series

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = z^2 + 1 + z
sage: f(0)
1
sage: f(L(0))
1
sage: f(f)
3 + 3*z + 4*z^2 + 2*z^3 + z^4
sage: g = z^-3/(1-2*z); g
z^-3 + 2*z^-2 + 4*z^-1 + 8 + 16*z + 32*z^2 + 64*z^3 + O(z^4)
sage: f(g)
z^-6 + 4*z^-5 + 12*z^-4 + 33*z^-3 + 82*z^-2 + 196*z^-1 + 457 + O(z)
sage: g^2 + 1 + g
z^-6 + 4*z^-5 + 12*z^-4 + 33*z^-3 + 82*z^-2 + 196*z^-1 + 457 + O(z)
sage: f(int(2))
7

sage: f = z^-2 + z + 4*z^3
sage: f(f)
4*z^-6 + 12*z^-3 + z^-2 + 48*z^-1 + 12 + O(z)
sage: f^-2 + f + 4*f^3
4*z^-6 + 12*z^-3 + z^-2 + 48*z^-1 + 12 + O(z)
sage: f(g)
4*z^-9 + 24*z^-8 + 96*z^-7 + 320*z^-6 + 960*z^-5 + 2688*z^-4 + 7169*z^-3 + O(z^-2)
sage: g^-2 + g + 4*g^3
4*z^-9 + 24*z^-8 + 96*z^-7 + 320*z^-6 + 960*z^-5 + 2688*z^-4 + 7169*z^-3 + O(z^-2)

sage: f = z^-3 + z^-2 + 1 / (1 + z^2); f
z^-3 + z^-2 + 1 - z^2 + O(z^4)
sage: g = z^3 / (1 + z - z^3); g
z^3 - z^4 + z^5 - z^7 + 2*z^8 - 2*z^9 + O(z^10)
sage: f(g)
z^-9 + 3*z^-8 + 3*z^-7 - z^-6 - 4*z^-5 - 2*z^-4 + z^-3 + O(z^-2)
sage: g^-3 + g^-2 + 1 / (1 + g^2)
z^-9 + 3*z^-8 + 3*z^-7 - z^-6 - 4*z^-5 - 2*z^-4 + z^-3 + O(z^-2)

sage: f = z^-3
sage: g = z^-2 + z^-1
sage: g^(-3)
z^6 - 3*z^7 + 6*z^8 - 10*z^9 + 15*z^10 - 21*z^11 + 28*z^12 + O(z^13)
sage: f(g)
z^6 - 3*z^7 + 6*z^8 - 10*z^9 + 15*z^10 - 21*z^11 + 28*z^12 + O(z^13)

sage: f = z^2 + z^3
sage: g = z^-3 + z^-2
sage: f^-3 + f^-2
z^-6 - 3*z^-5 + 7*z^-4 - 12*z^-3 + 18*z^-2 - 25*z^-1 + 33 + O(z)
sage: g(f)
z^-6 - 3*z^-5 + 7*z^-4 - 12*z^-3 + 18*z^-2 - 25*z^-1 + 33 + O(z)
sage: g^2 + g^3
z^-9 + 3*z^-8 + 3*z^-7 + 2*z^-6 + 2*z^-5 + z^-4
sage: f(g)
z^-9 + 3*z^-8 + 3*z^-7 + 2*z^-6 + 2*z^-5 + z^-4

sage: f = L(lambda n: n, valuation=0); f
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: f(z^2)
z^2 + 2*z^4 + 3*z^6 + O(z^7)

sage: f = L(lambda n: n, valuation=-2); f
-2*z^-2 - z^-1 + z + 2*z^2 + 3*z^3 + 4*z^4 + O(z^5)
sage: f3 = f(z^3); f3
-2*z^-6 - z^-3 + O(z)
sage: [f3[i] for i in range(-6,13)]
[-2, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4]

We compose a Laurent polynomial with a generic element:

sage: R.<x> = QQ[]
sage: f = z^2 + 1 + z^-1
sage: g = x^2 + x + 3
sage: f(g)
(x^6 + 3*x^5 + 12*x^4 + 19*x^3 + 37*x^2 + 28*x + 31)/(x^2 + x + 3)
sage: f(g) == g^2 + 1 + g^-1
True

We compose with another lazy Laurent series:

sage: LS.<y> = LazyLaurentSeriesRing(QQ)
sage: f = z^2 + 1 + z^-1
sage: fy = f(y); fy
y^-1 + 1 + y^2
sage: fy.parent() is LS
True
sage: g = y - y
sage: f(g)
Traceback (most recent call last):
...
ZeroDivisionError: the valuation of the series must be nonnegative

sage: g = 1 - y
sage: f(g)
3 - y + 2*y^2 + y^3 + y^4 + y^5 + O(y^6)
sage: g^2 + 1 + g^-1
3 - y + 2*y^2 + y^3 + y^4 + y^5 + O(y^6)

sage: f = L(lambda n: n, valuation=0); f
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: f(0)
0
sage: f(y)
y + 2*y^2 + 3*y^3 + 4*y^4 + 5*y^5 + 6*y^6 + O(y^7)
sage: fp = f(y - y)
sage: fp == 0
True
sage: fp.parent() is LS
True

sage: f = z^2 + 3 + z
sage: f(y - y)
3

With both of them sparse:

sage: L.<z> = LazyLaurentSeriesRing(QQ, sparse=True)
sage: LS.<y> = LazyLaurentSeriesRing(QQ, sparse=True)
sage: f = L(lambda n: 1, valuation=0); f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: f(y^2)
1 + y^2 + y^4 + y^6 + O(y^7)

sage: fp = f - 1 + z^-2; fp
z^-2 + z + z^2 + z^3 + z^4 + O(z^5)
sage: fpy = fp(y^2); fpy
y^-4 + y^2 + O(y^3)
sage: fpy.parent() is LS
True
sage: [fpy[i] for i in range(-4,11)]
[1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

sage: g = LS(valuation=2, constant=1); g
y^2 + y^3 + y^4 + O(y^5)
sage: fg = f(g); fg
1 + y^2 + y^3 + 2*y^4 + 3*y^5 + 5*y^6 + O(y^7)
sage: 1 + g + g^2 + g^3 + g^4 + g^5 + g^6
1 + y^2 + y^3 + 2*y^4 + 3*y^5 + 5*y^6 + O(y^7)

sage: h = LS(lambda n: 1 if n % 2 else 0, valuation=2); h
y^3 + y^5 + y^7 + O(y^9)
sage: fgh = fg(h); fgh
1 + y^6 + O(y^7)
sage: [fgh[i] for i in range(0, 15)]
[1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 6, 13]
sage: t = 1 + h^2 + h^3 + 2*h^4 + 3*h^5 + 5*h^6
sage: [t[i] for i in range(0, 15)]
[1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 6, 13]

We look at mixing the sparse and the dense:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: f = L(lambda n: 1, valuation=0); f
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: g = LS(lambda n: 1, valuation=1); g
y + y^2 + y^3 + y^4 + y^5 + y^6 + y^7 + O(y^8)
sage: f(g)
1 + y + 2*y^2 + 4*y^3 + 8*y^4 + 16*y^5 + 32*y^6 + O(y^7)

sage: f = z^-2 + 1 + z
sage: g = 1/(y*(1-y)); g
y^-1 + 1 + y + y^2 + y^3 + y^4 + y^5 + O(y^6)
sage: f(g)
y^-1 + 2 + y + 2*y^2 - y^3 + 2*y^4 + y^5 + O(y^6)
sage: g^-2 + 1 + g
y^-1 + 2 + y + 2*y^2 - y^3 + 2*y^4 + y^5 + O(y^6)

sage: f = z^-3 + z^-2 + 1
sage: g = 1/(y^2*(1-y)); g
y^-2 + y^-1 + 1 + y + y^2 + y^3 + y^4 + O(y^5)
sage: f(g)
1 + y^4 - 2*y^5 + 2*y^6 + O(y^7)
sage: g^-3 + g^-2 + 1
1 + y^4 - 2*y^5 + 2*y^6 + O(y^7)
sage: z(y)
y

We look at cases where the composition does not exist. \(g = 0\) and \(val(f) < 0\):

sage: g = L(0)
sage: f = z^-1 + z^-2
sage: f.valuation() < 0
True
sage: f(g)
Traceback (most recent call last):
...
ZeroDivisionError: the valuation of the series must be nonnegative

\(g \neq 0\) and \(val(g) \leq 0\) and \(f\) has infinitely many non-zero coefficients`:

sage: g = z^-1 + z^-2
sage: g.valuation() <= 0
True
sage: f = L(lambda n: n, valuation=0)
sage: f(g)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: f = L(lambda n: n, valuation=1)
sage: f(1 + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

We compose the exponential with a Dirichlet series:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: e = L(lambda n: 1/factorial(n), 0)
sage: D = LazyDirichletSeriesRing(QQ, "s")
sage: g = D(constant=1)-1; g
1/(2^s) + 1/(3^s) + 1/(4^s) + O(1/(5^s))

sage: e(g)[0:10]
[0, 1, 1, 1, 3/2, 1, 2, 1, 13/6, 3/2]

sage: sum(g^k/factorial(k) for k in range(10))[0:10]
[0, 1, 1, 1, 3/2, 1, 2, 1, 13/6, 3/2]

sage: g = D([0,1,0,1,1,2]); g
1/(2^s) + 1/(4^s) + 1/(5^s) + 2/6^s
sage: e(g)[0:10]
[0, 1, 1, 0, 3/2, 1, 2, 0, 7/6, 0]
sage: sum(g^k/factorial(k) for k in range(10))[0:10]
[0, 1, 1, 0, 3/2, 1, 2, 0, 7/6, 0]

sage: e(D([1,0,1]))
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: e5 = L(e, degree=5); e5
1 + z + 1/2*z^2 + 1/6*z^3 + 1/24*z^4
sage: e5(g)
1 + 1/(2^s) + 3/2/4^s + 1/(5^s) + 2/6^s + O(1/(8^s))
sage: sum(e5[k] * g^k for k in range(5))
1 + 1/(2^s) + 3/2/4^s + 1/(5^s) + 2/6^s + O(1/(8^s))

The output parent is always the common parent between the base ring of \(f\) and the parent of \(g\) or extended to the corresponding lazy series:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: R.<x> = ZZ[]
sage: parent(z(x))
Univariate Polynomial Ring in x over Rational Field
sage: parent(z(R.zero()))
Univariate Polynomial Ring in x over Rational Field
sage: parent(z(0))
Rational Field
sage: f = 1 / (1 - z)
sage: f(x)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + O(x^7)
sage: three = L(3)(x^2); three
3
sage: parent(three)
Univariate Polynomial Ring in x over Rational Field

polynomial(degree=None, name=None)

Return self as a Laurent polynomial if self is actually so.

INPUT:

• degree – None or an integer

• name – name of the variable; if it is None, the name of the variable of the series is used

OUTPUT:

A Laurent polynomial if the valuation of the series is negative or a polynomial otherwise.

If degree is not None, the terms of the series of degree greater than degree are truncated first. If degree is None and the series is not a polynomial or a Laurent polynomial, a ValueError is raised.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = L([1,0,0,2,0,0,0,3], valuation=5); f
z^5 + 2*z^8 + 3*z^12
sage: f.polynomial()
3*z^12 + 2*z^8 + z^5

revert()

Return the compositional inverse of self.

Given a Laurent Series \(f\). the compositional inverse is a Laurent Series \(g\) over the same base ring, such that \((f \circ g)(z) = f(g(z)) = z\).

The compositional inverse exists if and only if:

• \(val(f) = 1\), or

• \(f = a + b z\) with \(a b \neq 0\), or

• \(f = a/z\) with \(a \neq 0\)

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: z.revert()
z + O(z^8)
sage: (1/z).revert()
z^-1
sage: (z-z^2).revert()
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

class sage.rings.lazy_series.LazyModuleElement(parent, coeff_stream)

Bases: sage.structure.element.Element

A lazy sequence with a module structure given by term-wise addition and scalar multiplication.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: M = L(lambda n: n, valuation=0)
sage: N = L(lambda n: 1, valuation=0)
sage: M[:10]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: N[:10]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Two sequences can be added:

sage: O = M + N
sage: O[0:10]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Two sequences can be subtracted:

sage: P = M - N
sage: P[:10]
[-1, 0, 1, 2, 3, 4, 5, 6, 7, 8]

A sequence can be multiplied by a scalar:

sage: Q = 2 * M
sage: Q[:10]
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

The negation of a sequence can also be found:

sage: R = -M
sage: R[:10]
[0, -1, -2, -3, -4, -5, -6, -7, -8, -9]

arccos()

Return the arccos of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(RR)
sage: arccos(z)
1.57079632679490 - 1.00000000000000*z + 0.000000000000000*z^2
 - 0.166666666666667*z^3 + 0.000000000000000*z^4
 - 0.0750000000000000*z^5 + O(1.00000000000000*z^7)

sage: L.<z> = LazyLaurentSeriesRing(SR)
sage: arccos(z/(1-z))
1/2*pi - z - z^2 - 7/6*z^3 - 3/2*z^4 - 83/40*z^5 - 73/24*z^6 + O(z^7)

sage: L.<x, y> = LazyTaylorSeriesRing(SR)  # not tested
sage: arccos(x/(1-y))  # not tested
1/2*pi + (-x) + (-x*y) + ((-1/6)*x^3-x*y^2) + ((-1/2)*x^3*y-x*y^3)
 + ((-3/40)*x^5-x^3*y^2-x*y^4) + ((-3/8)*x^5*y+(-5/3)*x^3*y^3-x*y^5) + O(x,y)^7

arccot()

Return the arctangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(RR)
sage: arccot(z)
1.57079632679490 - 1.00000000000000*z + 0.000000000000000*z^2
 + 0.333333333333333*z^3 + 0.000000000000000*z^4
 - 0.200000000000000*z^5 + O(1.00000000000000*z^7)

sage: L.<z> = LazyLaurentSeriesRing(SR)
sage: arccot(z/(1-z))
1/2*pi - z - z^2 - 2/3*z^3 + 4/5*z^5 + 4/3*z^6 + O(z^7)

sage: L.<x, y> = LazyTaylorSeriesRing(SR)  # not tested
sage: acot(x/(1-y))  # not tested
1/2*pi + (-x) + (-x*y) + (1/3*x^3-x*y^2) + (x^3*y-x*y^3)
 + ((-1/5)*x^5+2*x^3*y^2-x*y^4) + (-x^5*y+10/3*x^3*y^3-x*y^5) + O(x,y)^7

arcsin()

Return the arcsin of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: arcsin(z)
z + 1/6*z^3 + 3/40*z^5 + 5/112*z^7 + O(z^8)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: asin(x/(1-y))  # not tested
x + x*y + (1/6*x^3+x*y^2) + (1/2*x^3*y+x*y^3)
 + (3/40*x^5+x^3*y^2+x*y^4) + (3/8*x^5*y+5/3*x^3*y^3+x*y^5)
 + (5/112*x^7+9/8*x^5*y^2+5/2*x^3*y^4+x*y^6) + O(x,y)^8

arcsinh()

Return the inverse of the hyperbolic sine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: asinh(z)
z - 1/6*z^3 + 3/40*z^5 - 5/112*z^7 + O(z^8)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: asinh(x/(1-y))  # not tested
x + x*y + ((-1/6)*x^3+x*y^2) + ((-1/2)*x^3*y+x*y^3)
 + (3/40*x^5-x^3*y^2+x*y^4) + (3/8*x^5*y+(-5/3)*x^3*y^3+x*y^5)
 + ((-5/112)*x^7+9/8*x^5*y^2+(-5/2)*x^3*y^4+x*y^6) + O(x,y)^8

arctan()

Return the arctangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: arctan(z)
z - 1/3*z^3 + 1/5*z^5 - 1/7*z^7 + O(z^8)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: atan(x/(1-y))  # not tested
x + x*y + ((-1/3)*x^3+x*y^2) + (-x^3*y+x*y^3)
 + (1/5*x^5+(-2)*x^3*y^2+x*y^4) + (x^5*y+(-10/3)*x^3*y^3+x*y^5)
 + ((-1/7)*x^7+3*x^5*y^2+(-5)*x^3*y^4+x*y^6) + O(x,y)^8

arctanh()

Return the inverse of the hyperbolic tangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: atanh(z)
z + 1/3*z^3 + 1/5*z^5 + 1/7*z^7 + O(z^8)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: atanh(x/(1-y))  # not tested
x + x*y + (1/3*x^3+x*y^2) + (x^3*y+x*y^3) + (1/5*x^5+2*x^3*y^2+x*y^4)
 + (x^5*y+10/3*x^3*y^3+x*y^5) + (1/7*x^7+3*x^5*y^2+5*x^3*y^4+x*y^6) + O(x,y)^8

change_ring(ring)

Return self with coefficients converted to elements of ring.

INPUT:

• ring – a ring

EXAMPLES:

Dense Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: s = 2 + z
sage: t = s.change_ring(QQ)
sage: t^-1
1/2 - 1/4*z + 1/8*z^2 - 1/16*z^3 + 1/32*z^4 - 1/64*z^5 + 1/128*z^6 + O(z^7)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: N = M.change_ring(QQ)
sage: N.parent()
Lazy Laurent Series Ring in z over Rational Field
sage: M.parent()
Lazy Laurent Series Ring in z over Integer Ring

Sparse Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: M.parent()
Lazy Laurent Series Ring in z over Integer Ring
sage: N = M.change_ring(QQ)
sage: N.parent()
Lazy Laurent Series Ring in z over Rational Field
sage: M^-1
z^-1 - 2 + z + O(z^6)

A Dirichlet series example:

sage: L = LazyDirichletSeriesRing(ZZ, 'z')
sage: s = L(constant=2)
sage: t = s.change_ring(QQ)
sage: t.parent()
Lazy Dirichlet Series Ring in z over Rational Field
sage: t^-1
1/2 - 1/2/2^z - 1/2/3^z - 1/2/5^z + 1/2/6^z - 1/2/7^z + O(1/(8^z))

coefficient(n)

Return the coefficient of the term with exponent n of the series.

INPUT:

• n – integer; the exponent

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: f = z / (1 - 2*z^3)
sage: [f[n] for n in range(20)]
[0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]
sage: f[0:20]
[0, 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64]

sage: M = L(lambda n: n, valuation=0)
sage: [M[n] for n in range(20)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: M = L(lambda n: n, valuation=0)
sage: [M[n] for n in range(20)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: f = L(lambda n: n)
sage: [f[n] for n in range(1, 11)]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
sage: f[1:11]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

sage: M = L(lambda n: n)
sage: [M[n] for n in range(1, 11)]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
sage: L = LazyDirichletSeriesRing(ZZ, "z", sparse=True)
sage: M = L(lambda n: n)
sage: [M[n] for n in range(1, 11)]
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

cos()

Return the cosine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: cos(z)
1 - 1/2*z^2 + 1/24*z^4 - 1/720*z^6 + O(z^7)

sage: L.<x,y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: cos(x/(1-y)).finite_part(4)  # not tested
1/24*x^4 + (-3/2)*x^2*y^2 - x^2*y + (-1/2)*x^2 + 1

cosh()

Return the cosh of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: cosh(z)
1 + 1/2*z^2 + 1/24*z^4 + 1/720*z^6 + O(z^7)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: cosh(x/(1-y))  # not tested
1 + 1/2*x^2 + x^2*y + (1/24*x^4+3/2*x^2*y^2) + (1/6*x^4*y+2*x^2*y^3)
 + (1/720*x^6+5/12*x^4*y^2+5/2*x^2*y^4) + O(x,y)^7

cot()

Return the cotangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: cot(z)
z^-1 - 1/3*z - 1/45*z^3 - 2/945*z^5 + O(z^6)

sage: L.<x> = LazyLaurentSeriesRing(QQ)
sage: cot(x/(1-x)).polynomial(4)
x^-1 - 1 - 1/3*x - 1/3*x^2 - 16/45*x^3 - 2/5*x^4

coth()

Return the hyperbolic cotangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: coth(z)
z^-1 + 1/3*z - 1/45*z^3 + 2/945*z^5 + O(z^6)

sage: coth(z + z^2)
z^-1 - 1 + 4/3*z - 2/3*z^2 + 44/45*z^3 - 16/15*z^4 + 884/945*z^5 + O(z^6)

csc()

Return the cosecant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: csc(z)
z^-1 + 1/6*z + 7/360*z^3 + 31/15120*z^5 + O(z^6)

sage: L.<x> = LazyLaurentSeriesRing(QQ)
sage: csc(x/(1-x)).polynomial(4)
x^-1 - 1 + 1/6*x + 1/6*x^2 + 67/360*x^3 + 9/40*x^4

csch()

Return the hyperbolic cosecant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: csch(z)
z^-1 - 1/6*z + 7/360*z^3 - 31/15120*z^5 + O(z^6)

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: csch(z/(1-z))
z^-1 - 1 - 1/6*z - 1/6*z^2 - 53/360*z^3 - 13/120*z^4 - 787/15120*z^5 + O(z^6)

define(s)

Define an equation by self = s.

INPUT:

• s – a Laurent polynomial

EXAMPLES:

We begin by constructing the Catalan numbers:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: C = L(None, valuation=0)
sage: C.define(1 + z*C^2)
sage: C
1 + z + 2*z^2 + 5*z^3 + 14*z^4 + 42*z^5 + 132*z^6 + O(z^7)

The Catalan numbers but with a valuation 1:

sage: B = L(None, valuation=1)
sage: B.define(z + B^2)
sage: B
z + z^2 + 2*z^3 + 5*z^4 + 14*z^5 + 42*z^6 + 132*z^7 + O(z^8)

We can define multiple series that are linked:

sage: s = L(None, valuation=0)
sage: t = L(None, valuation=0)
sage: s.define(1 + z*t^3)
sage: t.define(1 + z*s^2)
sage: s[:9]
[1, 1, 3, 9, 34, 132, 546, 2327, 10191]
sage: t[:9]
[1, 1, 2, 7, 24, 95, 386, 1641, 7150]

A bigger example:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: A = L(None, valuation=5)
sage: B = L(None, valuation=0)
sage: C = L(None, valuation=2)
sage: A.define(z^5 + B^2)
sage: B.define(z^5 + C^2)
sage: C.define(z^2 + C^2 + A^2)
sage: A[0:15]
[0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 5, 4, 14, 10, 48]
sage: B[0:15]
[0, 0, 0, 0, 1, 1, 2, 0, 5, 0, 14, 0, 44, 0, 138]
sage: C[0:15]
[0, 0, 1, 0, 1, 0, 2, 0, 5, 0, 15, 0, 44, 2, 142]

Counting binary trees:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: s = L(None, valuation=1)
sage: s.define(z + (s^2+s(z^2))/2)
sage: [s[i] for i in range(9)]
[0, 1, 1, 1, 2, 3, 6, 11, 23]

The \(q\)-Catalan numbers:

sage: R.<q> = ZZ[]
sage: L.<z> = LazyLaurentSeriesRing(R)
sage: s = L(None, valuation=0)
sage: s.define(1+z*s*s(q*z))
sage: s
1 + z + (q + 1)*z^2 + (q^3 + q^2 + 2*q + 1)*z^3
 + (q^6 + q^5 + 2*q^4 + 3*q^3 + 3*q^2 + 3*q + 1)*z^4
 + (q^10 + q^9 + 2*q^8 + 3*q^7 + 5*q^6 + 5*q^5 + 7*q^4 + 7*q^3 + 6*q^2 + 4*q + 1)*z^5
 + (q^15 + q^14 + 2*q^13 + 3*q^12 + 5*q^11 + 7*q^10 + 9*q^9 + 11*q^8
    + 14*q^7 + 16*q^6 + 16*q^5 + 17*q^4 + 14*q^3 + 10*q^2 + 5*q + 1)*z^6 + O(z^7)

We count unlabeled ordered trees by total number of nodes and number of internal nodes:

sage: R.<q> = QQ[]
sage: Q.<z> = LazyLaurentSeriesRing(R)
sage: leaf = z
sage: internal_node = q * z
sage: L = Q(constant=1, degree=1)
sage: T = Q(None, valuation=1)
sage: T.define(leaf + internal_node * L(T))
sage: [T[i] for i in range(6)]
[0, 1, q, q^2 + q, q^3 + 3*q^2 + q, q^4 + 6*q^3 + 6*q^2 + q]

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: g = L(constant=1, valuation=2)
sage: F = L(None); F.define(1 + g*F)
sage: [F[i] for i in range(1, 16)]
[1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3]
sage: oeis(_)                                                       # optional, internet
0: A002033: Number of perfect partitions of n.
1: A074206: Kalmár's [Kalmar's] problem: number of ordered factorizations of n.
...

sage: F = L(None); F.define(1 + g*F*F)
sage: [F[i] for i in range(1, 16)]
[1, 1, 1, 3, 1, 5, 1, 10, 3, 5, 1, 24, 1, 5, 5]

exp()

Return the exponential series of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: exp(z)
1 + z + 1/2*z^2 + 1/6*z^3 + 1/24*z^4 + 1/120*z^5 + 1/720*z^6 + O(z^7)
sage: exp(z + z^2)
1 + z + 3/2*z^2 + 7/6*z^3 + 25/24*z^4 + 27/40*z^5 + 331/720*z^6 + O(z^7)
sage: exp(0)
1
sage: exp(1 + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: L.<x,y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: exp(x+y)[4].factor()  # not tested
(1/24) * (x + y)^4
sage: exp(x/(1-y)).finite_part(3)  # not tested
1/6*x^3 + x^2*y + x*y^2 + 1/2*x^2 + x*y + x + 1

hypergeometric(a, b)

Return the \({}_{p}F_{q}\)-hypergeometric function \(\,_pF_{q}\) where \((p,q)\) is the parameterization of self.

INPUT:

• a – the first parameter of the hypergeometric function

• b – the second parameter of the hypergeometric function

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: z.hypergeometric([1, 1], [1])
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: z.hypergeometric([], []) - exp(z)
O(z^7)

log()

Return the series for the natural logarithm of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: log(1/(1-z))
z + 1/2*z^2 + 1/3*z^3 + 1/4*z^4 + 1/5*z^5 + 1/6*z^6 + 1/7*z^7 + O(z^8)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: log((1 + x/(1-y))).finite_part(3)  # not tested
1/3*x^3 - x^2*y + x*y^2 + (-1/2)*x^2 + x*y + x

map_coefficients(func, ring=None)

Return the series with func applied to each nonzero coefficient of self.

INPUT:

• func – function that takes in a coefficient and returns a new coefficient

EXAMPLES:

Dense Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: s = z/(1 - 2*z^2)
sage: t = s.map_coefficients(lambda c: c + 1)
sage: s
z + 2*z^3 + 4*z^5 + 8*z^7 + O(z^8)
sage: t
2*z + 3*z^3 + 5*z^5 + 9*z^7 + O(z^8)
sage: m = L(lambda n: n, valuation=0); m
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: m.map_coefficients(lambda c: c + 1)
2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)

Sparse Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: m = L(lambda n: n, valuation=0); m
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: m.map_coefficients(lambda c: c + 1)
2*z + 3*z^2 + 4*z^3 + 5*z^4 + 6*z^5 + 7*z^6 + O(z^7)

An example where the series is known to be exact:

sage: f = z + z^2 + z^3
sage: f.map_coefficients(lambda c: c + 1)
2*z + 2*z^2 + 2*z^3

Similarly for Dirichlet series:

sage: L = LazyDirichletSeriesRing(ZZ, "z")
sage: s = L(lambda n: n-1); s
1/(2^z) + 2/3^z + 3/4^z + 4/5^z + 5/6^z + 6/7^z + O(1/(8^z))
sage: s.map_coefficients(lambda c: c + 1)
2/2^z + 3/3^z + 4/4^z + 5/5^z + 6/6^z + 7/7^z + O(1/(8^z))

prec()

Return the precision of the series, which is infinity.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = 1/(1 - z)
sage: f.prec()
+Infinity

sec()

Return the secant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sec(z)
1 + 1/2*z^2 + 5/24*z^4 + 61/720*z^6 + O(z^7)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: sec(x/(1-y)).finite_part(4)  # not tested
5/24*x^4 + 3/2*x^2*y^2 + x^2*y + 1/2*x^2 + 1

sech()

Return the hyperbolic secant of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sech(z)
1 - 1/2*z^2 + 5/24*z^4 - 61/720*z^6 + O(z^7)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: sech(x/(1-y))  # not tested
1 + ((-1/2)*x^2) + (-x^2*y) + (5/24*x^4+(-3/2)*x^2*y^2)
 + (5/6*x^4*y+(-2)*x^2*y^3) + ((-61/720)*x^6+25/12*x^4*y^2+(-5/2)*x^2*y^4)
 + O(x,y)^7

shift(n)

Return self with the indices shifted by n.

For example, a Laurent series is multiplied by the power \(z^n\), where \(z\) is the variable of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(ZZ)
sage: f = 1 / (1 + 2*z)
sage: f
1 - 2*z + 4*z^2 - 8*z^3 + 16*z^4 - 32*z^5 + 64*z^6 + O(z^7)
sage: f.shift(3)
z^3 - 2*z^4 + 4*z^5 - 8*z^6 + 16*z^7 - 32*z^8 + 64*z^9 + O(z^10)
sage: f << -3  # shorthand
z^-3 - 2*z^-2 + 4*z^-1 - 8 + 16*z - 32*z^2 + 64*z^3 + O(z^4)
sage: g = z^-3 + 3 + z^2
sage: g.shift(5)
z^2 + 3*z^5 + z^7
sage: L([2,0,3], valuation=2, degree=7, constant=1) << -2
2 + 3*z^2 + z^5 + z^6 + z^7 + O(z^8)

sage: D = LazyDirichletSeriesRing(QQ, 't')
sage: f = D([0,1,2]); f
1/(2^t) + 2/3^t
sage: f.shift(3)
1/(5^t) + 2/6^t

sin()

Return the sine of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sin(z)
z - 1/6*z^3 + 1/120*z^5 - 1/5040*z^7 + O(z^8)

sage: sin(1 + z)
Traceback (most recent call last):
...
ValueError: can only compose with a positive valuation series

sage: L.<x,y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: sin(x/(1-y)).finite_part(3)  # not tested
(-1/6)*x^3 + x*y^2 + x*y + x

sinh()

Return the sinh of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sinh(z)
z + 1/6*z^3 + 1/120*z^5 + 1/5040*z^7 + O(z^8)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: sinh(x/(1-y))  # not tested
x + x*y + (1/6*x^3+x*y^2) + (1/2*x^3*y+x*y^3)
 + (1/120*x^5+x^3*y^2+x*y^4) + (1/24*x^5*y+5/3*x^3*y^3+x*y^5)
 + (1/5040*x^7+1/8*x^5*y^2+5/2*x^3*y^4+x*y^6) + O(x,y)^8

sqrt()

Return self^(1/2).

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: sqrt(1+z)
1 + 1/2*z - 1/8*z^2 + 1/16*z^3 - 5/128*z^4 + 7/256*z^5 - 21/1024*z^6 + O(z^7)

sage: L.<x,y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: sqrt(1+x/(1-y))  # not tested
1 + 1/2*x + ((-1/8)*x^2+1/2*x*y) + (1/16*x^3+(-1/4)*x^2*y+1/2*x*y^2)
 + ((-5/128)*x^4+3/16*x^3*y+(-3/8)*x^2*y^2+1/2*x*y^3)
 + (7/256*x^5+(-5/32)*x^4*y+3/8*x^3*y^2+(-1/2)*x^2*y^3+1/2*x*y^4)
 + ((-21/1024)*x^6+35/256*x^5*y+(-25/64)*x^4*y^2+5/8*x^3*y^3+(-5/8)*x^2*y^4+1/2*x*y^5)
 + O(x,y)^7

This also works for Dirichlet series:

sage: D = LazyDirichletSeriesRing(SR, "s")
sage: Z = D(constant=1)
sage: f = sqrt(Z)
sage: f
1 + 1/2/2^s + 1/2/3^s + 3/8/4^s + 1/2/5^s + 1/4/6^s + 1/2/7^s + O(1/(8^s))
sage: f*f - Z
O(1/(8^s))

tan()

Return the tangent of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ) sage: tan(z) z + 1/3*z^3 + 2/15*z^5 + 17/315*z^7 + O(z^8)

sage: L.<x,y> = LazyTaylorSeriesRing(QQ) # not tested sage: tan(x/(1-y)).finite_part(5) # not tested 2/15*x^5 + 2*x^3*y^2 + x*y^4 + x^3*y + x*y^3 + 1/3*x^3 + x*y^2 + x*y + x

tanh()

Return the tanh of self.

EXAMPLES:

sage: L.<z> = LazyLaurentSeriesRing(QQ)
sage: tanh(z)
z - 1/3*z^3 + 2/15*z^5 - 17/315*z^7 + O(z^8)

sage: L.<x, y> = LazyTaylorSeriesRing(QQ)  # not tested
sage: tanh(x/(1-y))  # not tested
x + x*y + ((-1/3)*x^3+x*y^2) + (-x^3*y+x*y^3)
 + (2/15*x^5+(-2)*x^3*y^2+x*y^4) + (2/3*x^5*y+(-10/3)*x^3*y^3+x*y^5)
 + ((-17/315)*x^7+2*x^5*y^2+(-5)*x^3*y^4+x*y^6) + O(x,y)^8

truncate(d)

Return this series with its terms of degree >= d truncated.

INPUT:

• d – integer; the degree from which the series is truncated

EXAMPLES:

Dense Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=False)
sage: alpha = 1/(1-z)
sage: alpha
1 + z + z^2 + z^3 + z^4 + z^5 + z^6 + O(z^7)
sage: beta = alpha.truncate(5)
sage: beta
1 + z + z^2 + z^3 + z^4
sage: alpha - beta
z^5 + z^6 + O(z^7)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: M.truncate(4)
z + 2*z^2 + 3*z^3

Sparse Implementation:

sage: L.<z> = LazyLaurentSeriesRing(ZZ, sparse=True)
sage: M = L(lambda n: n, valuation=0); M
z + 2*z^2 + 3*z^3 + 4*z^4 + 5*z^5 + 6*z^6 + O(z^7)
sage: M.truncate(4)
z + 2*z^2 + 3*z^3

Series which are known to be exact can also be truncated:

sage: M = z + z^2 + z^3 + z^4
sage: M.truncate(4)
z + z^2 + z^3