AngouriMath
OverTheRoots(AngouriMath.Functions.RationalPolynomial,System.Int32,AngouriMath.Functions.RationalPolynomial,AngouriMath.Functions.RationalPolynomial,AngouriMath.Functions.RationalPolynomial,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
The logarithms at the poles whose residues are roots of residues ,
as a sum over those poles:sum(A(r)/D'(r) ln(x - r), r in { r : E(r) = 0 }) ,
whereE is the factor of D they are the roots of. null where E does not have the roots the residues' multiplicities
count.
as a sum over those poles:
where
count.
Remarks
The residue at a simple pole b of A/D is A(b)/D'(b) , a root of the
resultant, andD'(b) is not zero there. So the poles whose residues are roots of
R are the common roots of D and the polynomial D'^n R(A/D') , and
E is their gcd over the rationals: no arithmetic in the field the residues lie in
is needed. The sum is the one Rothstein–Trager's theorem writes over the roots of the
resultant, with its terms taken pole by pole.E has a factor of degree above two,
since a pole in a field of degree at most two has its residue there too, so the sum is
left standing rather than written out in radicals, unless each such factor is a
binomial, whose roots are written.
https://github.com/asc-community/AngouriMath/issues/1285
resultant, and
is needed. The sum is the one Rothstein–Trager's theorem writes over the roots of the
resultant, with its terms taken pole by pole.
since a pole in a field of degree at most two has its residue there too, so the sum is
left standing rather than written out in radicals, unless each such factor is a
binomial, whose roots are written.
https://github.com/asc-community/AngouriMath/issues/1285
Angouri © 2019-2023 · Project's repo · Site's repo · Octicons · Transparency · 4378 pages online