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OverTheGaussianRoots(AngouriMath.Entity,System.Collections.Generic.List{System.ValueTuple{AngouriMath.Entity,System.Int32}},AngouriMath.Entity,System.Collections.Generic.Dictionary{PeterO.Numbers.EInteger,AngouriMath.Entity},System.Int32,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

The numerators p + q x over l^(k - m) Q^m, l the leading
coefficient and k the multiplicity, for m from one to
k at index m - 1, of the block at
Q^multiplicity -- the written factor written -- of
numerator over the product of factors, where
Q is a quadratic over the rationals with the roots r = ρ + σ i and its
conjugate, ρ and σ rational. Null where Q is not such a
quadratic, or a side does not read as a polynomial over the rationals in the symbols.

Remarks

The residue formula at the root: with Q^k = l^k (x - r)^k (x - r̄)^k, the
coefficient c_j of 1/(x - r)^j is the (k - j)th Taylor coefficient
at r of N/(R l^k (x - r̄)^k), R the other factors, which is the
quotient of two series in x - r -- one division per order. The coefficients
at r̄ are the conjugates: conjugation fixes the rationals in the symbols and so
fixes N/D, which holds as polynomial identities whatever the symbols are. So
the block is Σ 2 Re(c_j (x - r̄)^j)/M^j for the monic M = Q/l, and each
numerator, written in powers of M, gives its digits to the powers below it.
Over the Gaussian rationals as pairs of polynomials, the unit a variable j with j^2 = -1 read off as the parity of its power: every order of the series
is a pair over a power of the norm of the series' leading term, and nothing is put in
lowest terms until the last coefficient. The digits of (x - r̄)^j in powers of
M are numbers.
https://github.com/asc-community/AngouriMath/issues/233

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