AngouriMath
WantsASumOverRootsAlone(AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
Whether the logarithms of the integral of numerator over
denominator , a proper fraction of polynomials with rational
coefficients that Integrate(AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity.Variable) reads, have a residue in a field of degree
above two, so that Integrate(AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity.Variable) writes them only as a sum over roots, and
no product of the denominator's irreducible factors past the second degree is a
polynomial in(x + s)^k with k at least two.
coefficients that Integrate(AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity.Variable) reads, have a residue in a field of degree
above two, so that Integrate(AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity.Variable) writes them only as a sum over roots, and
no product of the denominator's irreducible factors past the second degree is a
polynomial in
Remarks
The residues of a rational function are its own, whatever spelling or split it is asked
in, and every antiderivative has them as the coefficients of its logarithms. The rules
that write such a residue in closed form read a block of the denominator that is a
binomial, a biquadratic or a polynomial in a power ofx , at its roots of unity or
in nested square roots, after the shift that takes away its second term where there is
one; each is a polynomial in(x + s)^k , and no product of the denominator's
irreducible factors may be one. The integral is then nothing the first pass writes,
which keeps sums over roots for the second.
in, and every antiderivative has them as the coefficients of its logarithms. The rules
that write such a residue in closed form read a block of the denominator that is a
binomial, a biquadratic or a polynomial in a power of
in nested square roots, after the shift that takes away its second term where there is
one; each is a polynomial in
irreducible factors may be one. The integral is then nothing the first pass writes,
which keeps sums over roots for the second.
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