#5226 csign(gamma(z)) is pnz for a z declared complex

[David Scherfgen]

Bug report written by Claude.

csign answers complex for sin(z), exp(z), log(z) or z^2 when z is declared complex, but pnz, real of unknown sign, for gamma(z), tan(z), erfc(z), zeta(z), bessel_j(0, z), psi[0](z), expintegral_e1(z), lambert_w(z), floor(z), an undefined f(z) and others. The abs rules of simpexpt and timesin in src/simp.lisp are guarded by csign and trust it:

(%i1) display2d : false$
(%i2) declare(z, complex)$
(%i3) map(csign, [sin(z), gamma(z), tan(z), erfc(z), f(z)]);
(%o3) [complex,pnz,pnz,pnz,pnz]
(%i4) [abs(gamma(z))^3, abs(gamma(z))^(2/3), gamma(z)^2/abs(gamma(z))^3, abs(f(z))^3];
(%o4) [gamma(z)^2*abs(gamma(z)),gamma(z)^(2/3),1/abs(gamma(z)),f(z)^2*abs(f(z))]

All four are wrong for a complex value: at z = -1.3+0.7*%i, abs(gamma(z))^3 is 0.311 and gamma(z)^2*abs(gamma(z)) is -0.158+0.268*%i.

The gap is in src/compar.lisp: an application whose operator has no sign rule of its own is treated as real, whatever its arguments. Answering complex there whenever an argument is not real by csign would close it; the functions that are real for a real argument, gamma and erfc among them, would keep pnz for a real one. Nothing changes for an undeclared symbol.