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PartialFractions

Description

Summary

One step of a partial fraction decomposition, at a coprime pair of factors of the
denominator rather than at a root of it.

Remarks

The sibling step, TrySplitOffRationalRoot(AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity.Variable,AngouriMath.Entity@,AngouriMath.Entity@,AngouriMath.Entity@), splits at a
rational root, which is all the decomposition there was: a denominator with no rational
root was left whole, so 1/(x^4 + 3x^2 + 2) had no antiderivative even though it
is (x^2 + 1)(x^2 + 2) and both of those are integrated by the rule for a linear
numerator over a quadratic. Nothing was missing but the split.
https://github.com/asc-community/AngouriMath/issues/919
One step and not the whole decomposition, for the same reason as the sibling: what comes
out is two strictly smaller problems of the same kind, and the integrator recurses into
them. Splitting D into coprime A and B, the extended Euclidean
algorithm gives U*A + V*B = 1, so N = N*V*B + N*U*A and
N/(A*B) = N*V/A + N*U/B. Each numerator is then reduced modulo its own
denominator; the polynomial parts that come off cannot survive, since a proper fraction
minus two proper fractions is a polynomial that vanishes at infinity.
No condition is owed.A and B being coprime, A*B is zero
exactly where one of them is, so the two sides are undefined at the same points and the
domain neither widens nor narrows. That is what makes this different from cancelling a
shared factor, which is where a decomposition usually loses a singularity.
The decomposition is produced only where every piece of it is a shape an integration
rule reads — see the guard below, which is what keeps declining cheap. A denominator
that is a power of one irreducible is declined for the further reason that it has no
coprime pair to split into at all. The ladder that decomposes that one — N/f^k as terms over f^k, f^(k-1), ... — is deliberately not built here: for
f linear the sibling step already does it, and for f quadratic every term
it produces is over (x^2 + c)^k, which nothing reads, so the decomposition would
end in the same unevaluated integral it started from.

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PartialFractions

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