AngouriMath
SplitRootsOfQuotientsIn(AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
Every root of a quotient of polynomials in expr written as a
quotient of roots, where that is exact:sqrt(P/Q) = sqrt(P)/sqrt(Q) on the
principal branch unlessQ < 0 < P , where the left is i sqrt(P/|Q|) and the right -i sqrt(P/|Q|) -- with both negative the two i s cancel and
withP negative alone they agree. So the rule asks whether there is a real
x with Q(x) < 0 < P(x) , on one point of each interval between the
real roots, and splits only where there is not.
quotient of roots, where that is exact:
principal branch unless
with
real roots, and splits only where there is not.
Remarks
Charlwood's arcsin(x/sqrt(1 - x^2)) by parts against one leaves
x (1 - x^2)^(-3/2)/sqrt((1 - 2x^2)/(1 - x^2)) , a root of a quotient that no
rule reads; split, it isx/((1 - x^2) sqrt(1 - 2x^2)) , which Euler answers.
1 - x^2 < 0 only past 1 , where 1 - 2x^2 is negative too.
rule reads; split, it is
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