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SolveOverTheComplexNumbers(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

The complex x at which expr is zero, where it is
made by arithmetic and whole powers from x, rational numbers, and
moduli, signums, conjugates, real and imaginary parts of such expressions;
null otherwise, or where the roots are not finitely many or not
found exactly.

Remarks

With x = a + b i for real a and b, and each modulus |u| a
real t, not negative, with t^2 = (Re u)^2 + (Im u)^2, the real and the
imaginary part of the equation and one equation for each modulus are polynomial
equations in as many unknowns, solved exactly by a Groebner basis. The roots are the
solutions with a, b and every t real and no t negative.
x |x| = 4 is a t = 4, b t = 0 and t^2 = a^2 + b^2, so
t = 2 and x = 2; and x^2 + 1 = |x| has the roots +-0.618 i,
which no split by the sign of x finds.
https://github.com/asc-community/AngouriMath/issues/2139
A signum sgn(u) is u / |u| where u is not zero and 0 where
it is, so each is solved in both cases: x + sgn(x) = 3 is { 2 } where
x is not zero, and has no root where it is. An argument is not taken, since it
is not a polynomial in the parts.

AngouriMath

Table of contents

AngouriMath

SolveOverTheComplexNumbers​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable)

SolveOverTheComplexNumbers(AngouriMath.Entity,AngouriMath.Entity.Variable)