Multiplicity-Weighted Moments Across a Harmonic Caustic: Closed Forms with Order-Independent Breakpoints

Abstract

While the Jacobian of a planar harmonic mapping keeps a constant sign, the oriented and the multiplicity-weighted surface elements agree up to that sign, and the two natural moment hierarchies coincide. The nearest classification results in the literature are stated under exactly this hypothesis; this paper works past it. Once the Jacobian changes sign, the critical circle acquires a caustic as its image, part of the plane is covered several times with opposite orientations, and the hierarchies part company: integrating a radial weight against the oriented surface element $n_z$ lets overlapping sheets cancel, giving the degree-weighted moments $J_n$, while integrating against $|n_z|$ lets every sheet count positively, giving the multiplicity-weighted moments $J_n^{\pm}$. These are the classical degree and area formulas for the same map, the second weighted by the multiplicity function (Banach indicatrix). It is the second family, the one the caustic threatens, for which a closed form is no longer automatic. Our main result is that it has one anyway: $J_n^{\pm}$ is the total variation of the very same explicit primitive whose endpoint difference gives $J_n$. Crossing the caustic therefore costs only finitely many extra evaluation points, and they are the same points for every order $n$; for a monotone profile there is at most one interior breakpoint, at the critical radius itself. The family carrying this is the two-monomial planar harmonic mapping $F(w)=w+iC\bar w^k$, $w=p(u)e^{iv}$, arising as the planar projection of a Fourier-perturbed surface. On the degree-weighted side $J_n$ has a boundary-only closed form for every $n\ge 0$, the zeroth being the oriented $z$-flux. Neither weighted integral is a new object, and the caustic has been located before in the literature on harmonic trinomials; what is new is the closed evaluation of the multiplicity-weighted hierarchy across it.