Note that the definition of equality for series not only affects the result of the application of eq and ne, but also that of IsOne, IsZero and IsMinusOne.
Given a series f, return whether f is weakly zero, which is whether f is exactly zero or of the form O(xp) for some p.
Given series f and g, return whether f is weakly equal to g, which is whether (f - g) is weakly zero (see IsWeaklyZero).
Given series f and g, return whether f is identical to g, which is whether f and g have exactly the same valuation, precision, and coefficients.
Given a series f, this returns the absolute precision that is stored with f. If f is a series in x, the absolute precision of f is the exponent p such that xp is the first term of f of which the coefficient is not known, that is, it is the least p such that f∈O(xp). If f is known exactly (in a free ring), the absolute precision is infinite and an error occurs. Note that the absolute precision may be a non-integral rational number if f is a Puiseux series.
Given a series f, this returns the relative precision that is stored with f. The relative precision counts the number of coefficients of f that is known, starting at the first non-zero term. Hence the relative precision is the difference between the absolute precision and the valuation of f, and is therefore always non-negative; however, if f is exact, the relative precision is infinite and the value ∞ is returned. Note that the relative precision may be a non-integral rational number if f is a Puiseux series.
The (non puiseux) series f with absolute precision r (which can be positive infinity).
Let f be a series with coefficients in a ring R and with indeterminate x. This function returns the sequence Q of coefficients of f, the unscaled valuation v and the exponent denominator d of f (v is the true valuation of f multiplied by d). The i-th entry Q[i] of Q equals the coefficient of x^((v + i - 1)/(d)) in f. Thus the first entry of Q is the `first' (lowest order) non-zero coefficient of f, i.e., the coefficient of xw where w is the true valuation of f.
Given a series f with coefficients in a ring R, and a rational or integer i, return the coefficient of the i-th power of the indeterminate x of f as an element of R. If f is a Puiseux series i may be a (non-integral) rational; otherwise i must be an integer (and also must be non-negative if f is a power series). Also, i must be less than p, the precision of f.
Given a series f with coefficients in a ring R, return the leading coefficient of f as an element of R, which is the first non-zero coefficient of f (i.e., the coefficient xv in f, where x is the indeterminate of f and v is the valuation of f).
Given a series f with coefficients in a ring R, return the leading term of f, which is the first non-zero term of f (i.e., the term of f whose monomial is xv, where x is the indeterminate of f and v is the valuation of f).
Given a series f, return the exact series obtained by truncating f after the last known non-zero coefficient.
Given a series f, return the exponent denominator of f, i.e., the lowest common denominator of all the exponents of the non-zero terms of f (always an integer). For power series and Laurent series, this will always be 1 of course.
Given a series f, return the degree of the truncation of f, that is, the exponent of the last known non-zero term. Note that this may be a non-integral rational number if f is a Puiseux series.
Given a series f, return the smallest integer v (possibly negative for Laurent series) such that the coefficient of xv in f is not known to be zero. For the exact 0 element (in a free ring), the valuation is ∞. Note that the valuation may be a non-integral rational number if f is a Puiseux series.
The exponent denominator of the series f. This is the lowest common denominator of the exponents of the non-zero terms of f.
Given a series f∈R, return the derivative of f with respect to its indeterminate, as an element of R. Note that the precision decreases by 1 (unless f has infinite precision).
Given a series f∈R and an integer n > 0, return the n-th derivative of f with respect to its indeterminate, as an element of R. Note that the precision decreases by n (unless f has infinite precision).
Given a series f∈R, return an anti-derivative F of f with respect to its indeterminate, which is an element of R which has derivative f. The coefficient of x - 1 in f must be zero. Note that the precision of F will be exceeding that of f by 1 (unless f has infinite precision).
Given an element f of a series ring over the coefficient ring R, and an element s of the ring S, return the value of f(s) when the indeterminate x is evaluated at s. The result will be an element of the common overstructure over R and S.
The Laplace transform of the series f; if f has expansion ∑i≥0 aixi, its Laplace transform has expansion ∑i≥0 (i!ai)xi. The valuation of f must be integral and non-negative.
Return the square root of the series f, f must have even valuation if it is a power or Laurent series.
Given elements f and g from the same series ring P, return their composition, defined by f g = ∑i < pfi(gi), where f=∑i < p fixi.
Given a series f (in x, say), this returns the inverse of f under composition, that is, an element g of the same power series ring such that its composition with f equals x to the best possible precision. If f is a power or Laurent series, the valuation of f must be 1. If f is a Puiseux series, the valuation of f must be positive (but need not equal 1), and if the valuation of f is not 1, the leading coefficient of f must be 1.
Given elements f and g from the same series ring P, return their convolution f ast g, defined by f ast g = ∑i < min(p, q)figixi, where f=∑i < p fixi + O(xp) and g=∑i < qgixi + O(xq).
> S<x> := PowerSeriesRing(RationalField());
> f := Sin(x);
> g := Arcsin(x);
> f;
x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 -
1/39916800*x^11 + 1/6227020800*x^13 - 1/1307674368000*x^15 +
1/355687428096000*x^17 - 1/121645100408832000*x^19 + O(x^21)
> g;
x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 +
231/13312*x^13 + 143/10240*x^15 + 6435/557056*x^17 +
12155/1245184*x^19 + O(x^21)
> Composition(f, g);
x + O(x^21)
> Composition(g, f);
x + O(x^21)
> Reversion(f) - g;
O(x^21)
> Reversion(g) - f;
O(x^21)
Next we compute the reversion of a series whose valuation is not 1.
> S<x> := PuiseuxSeriesRing(RationalField());
> f := x^3 - x^5 + 2*x^8;
> r := Reversion(f);
> f;
x^3 - x^5 + 2*x^8
> r;
x^(1/3) + 1/3*x + 4/9*x^(5/3) - 2/3*x^2 + 65/81*x^(7/3) -
22/9*x^(8/3) + 5/3*x^3 - 208/27*x^(10/3) + 5005/729*x^(11/3)
- 70/3*x^4 + 206264/6561*x^(13/3) - 50830/729*x^(14/3) +
134*x^5 - 498674/2187*x^(16/3) + 31389020/59049*x^(17/3) +
O(x^6)
> Composition(r, f);
x + O(x^18)
> Composition(f, r);
x + O(x^(20/3))
Finally we compute the reversion of a proper Puiseux series.
> f := x^(2/5) - x^(2/3) + x^(3/2) + O(x^2);
> r := Reversion(f);
> r;
x^(5/2) + 5/2*x^(19/6) + 145/24*x^(23/6) + 715/48*x^(9/2) +
389795/10368*x^(31/6) - 5/2*x^(21/4) + O(x^(11/2))
> Composition(f, r);
x + O(x^4)
> Composition(r, f);
x + O(x^(11/5))