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 (power or Laurent) 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 x^p is the first term of f of which the coefficient is not known, that is, it is the least p such that f in O(x^p). If f is known exactly (in a free ring), the absolute precision is infinite and an error occurs.
Given a (power or Laurent) 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 -1 is returned.
The coefficients of the series f in R[[X]] or f in R((X)) in ascending order, as a sequence of elements of R. The first entry in the sequence is the `first' (lowest order) non-zero coefficient of f, that is the coefficient of x^v where v is the valuation of f, while the last coefficient is the `last' non-zero coefficient. In particular the sequence of coefficients of 0 will be empty.
Given a series f in R[[X]] or f in R((X)), and an integer i, return the coefficient of the i-th power of the indeterminate X of f. This coefficient is an element of R. The integer i must be less than p, the precision of f; if f is a power series, i must be non-negative.
Given a power series f, return the exact power series obtained by truncating f after the last known non-zero coefficient.
Given a power series f, return the degree of the truncation of f, that is, the exponent of the last known non-zero term.
Given a series f in R[[X]] or f in R((X)), return the smallest integer v (possibly negative for Laurent series) such that the coefficient of X^v in f is not known to be zero. An exception is made for the exact 0 element (in a free ring), which will have valuation 0 (rather than infinity).
Returns the approximated square root of the power series f, if it has even valuation.
Given a power series f in 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 power series f in 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 power series f in R, return an anti-derivative F of f with respect to its indeterminate, which is an element of R which has derivative f. 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 power series f; if f has expansion sum_(i >= 0) a_ix^i, its Laplace transform has expansion sum_(i >= 0) (i!a_i)x^i.
Given elements f and g from the same power series ring P, return their composition, defined by f o g = sum_(i < p)f_i(g^i), where f=sum_(i < p) f_ix^i.
Given a power 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. Here f must have valuation 1, and may not be exact.
Given elements f and g from the same power series ring P, return their convolution f ast g, defined by f ast g = sum_(i < min(p, q))f_ig_ix^i, where f=sum_(i < p) f_ix^i + O(x^p) and g=sum_(i < q)g_ix^i + O(x^q).
In each of the functions below, the precision of the result
will be approximately equal to the precision of the argument.
An error will result if the coefficient ring of the power series is not a field.
If the first argument has a non-zero constant term, an error
will result unless the coefficient ring of the parent is a real
or complex domain (so that the transcendental function can be
evaluated in the constant term).
Sin(f) : RngSerElt -> RngSerElt
Returns the sine of the power series f defined over a field.
Returns the cosine of the power series f defined over a field.
Returns the tangent of the power series f defined over a field.
Returns the hyperbolic sine of the power series f defined over a field.
Returns the hyperbolic cosine of the power series f defined over a field.
Returns the hyperbolic tangent of the power series f defined over a field.
Returns the exponential of the power series f defined over a field.
Returns the logarithm of the power series f defined over a field. The valuation of f must be zero.[Next] [Prev] [_____] [Left] [Up] [Index] [Root]