A truncated power series is represented as a (dense) vector of coefficients. The precision (maximum number of coefficients) is statically checked. For instance, adding two series of different precision would result in a type error.

Create a polynomial with the given list of coefficients. E.g.

>>> (polynomial (Proxy :: Proxy 4) [1,1])^2
series (Proxy :: Proxy 4) [Val (1 % 1),Val (2 % 1),Val (1 % 1),Val (0 % 1)]

Create a power series with the given list of initial coefficients. E.g.

>>> (series (Proxy :: Proxy 4) [1,1])^2
series (Proxy :: Proxy 4) [Val (1 % 1),Val (2 % 1),Indet,Indet]

Create the power series x^k.

Create an empty power series. All its coefficients are Indet.

A series whose constant term is DZ.

If f :: Series n, then precision f = n.

The underlying vector of coefficients. E.g.

>>> coeffVector $ polynomial (Proxy :: Proxy 3) [1,2]
fromList [Val (1 % 1),Val (2 % 1),Val (0 % 1)]

The list of coefficients of the given series. E.g.

>>> coeffList $ series (Proxy :: Proxy 3) [9]
[Val (9 % 1),Indet,Indet]

Constant term of the given series.

The first nonzero coefficient when read from smaller to larger powers of x. If no such coefficient exists then return 0.

The longest initial segment of coefficients that are rational (i.e. not Indet or DZ).

The longest initial segment of coefficients that are integral

The longest initial segment of coefficients that are Ints

Evaluate the polynomial p at x using Horner's method.

>>> let x = polynomial (Proxy :: Proxy 5) [0,1]
>>> eval (1-x+x^2) 2
Val (3 % 1)

Coefficient wise multiplication of two power series. Also called the Hadamard product.

Coefficient wise division of two power series.

The power operator for Rats. E.g.

>>> (1/4) !^! (3/2)
Val (1 % 8)

A power series raised to a rational power.

>>> series (Proxy :: Proxy 4) [1,2,3,4] ^! (1/2)
series (Proxy :: Proxy 4) [Val (1 % 1),Val (1 % 1),Val (1 % 1),Val (1 % 1)]

Select certain coefficients of the first series, based on indices from the second series, returning the selection as a series. Elements of the second series that are not nonnegative integers or not less than the precision are ignored; trailing zeros are also ignored.

The composition of two power series.

>>> let x = polynomial (Proxy :: Proxy 4) [0,1]
>>> (1/(1-x)) `o` (2*x)
series (Proxy :: Proxy 4) [Val (1 % 1),Val (2 % 1),Val (4 % 1),Val (8 % 1)]

The (formal) derivative of a power series.

The (formal) integral of a power series.

Reversion (compositional inverse) of a power series. Unless the constant of the power series is zero the result is indeterminate.

>>> revert $ series (Proxy :: Proxy 4) [0,1,2,3]
series (Proxy :: Proxy 4) [Val (0 % 1),Val (1 % 1),Val ((-2) % 1),Val (5 % 1)]

>>> revert $ series (Proxy :: Proxy 4) [1,1,1,1]
series (Proxy :: Proxy 4) [Indet,Indet,Indet,Indet]

The secant function: sec f = 1 / cos f

The factorial of a constant power series. If the the power series isn't constant, then the result is indeterminate (represented using Indet).

>>> fac (polynomial (Proxy :: Proxy 4) [3])
series (Proxy :: Proxy 4) [Val (6 % 1),Val (0 % 1),Val (0 % 1),Val (0 % 1)]

Construct a series that has coefficient 1 for each term whose power is some coefficient of the input series and 0 elsewhere. Elements of the input series that are not nonnegative integers or not less than the precision are ignored; trailing zeros are also ignored.

>>> rseq $ series (Proxy :: Proxy 4) [1,3]
series (Proxy :: Proxy 4) [Val (0 % 1),Val (1 % 1),Val (0 % 1),Val (1 % 1)]

The "complement" of rseq, i.e., the series generated by rseq, with 0 replaced by 1, and 1 replaced by 0.