On resolution to Wu's Conjecture

TL;DR

This paper addresses Wu's conjecture by developing analytical methods to determine the singularity distribution of Cauchy's function outside a domain from boundary data, solving an inverse problem with broad applications.

Contribution

It introduces new analytical techniques, including complex algebra, asymptotic methods, and generalized Hilbert transforms, to resolve Wu's conjecture for various singularity types.

Findings

Successfully determines singularity distributions from boundary data.

Develops a general asymptotic method for multiple singularities.

Extends methods to cases with conjugate boundary functions.

Abstract

In this series of studies on Cauchy's function $f(z)$ ($z=x+iy$) and its integral $J[f(z)]\equiv (2\pi i)^{-1}\oint_C f(t)dt/(t-z)$ taken along a Jordan contour $C$, the aim is to investigate their comprehensive properties over the entire $z$-plane consisted of the simply-connected closed domain ${\cal D}^+$ bounded by $C$ and the open domain ${\cal D}^-$ outside $C$. This article attempts to solve an inverse problem that Cauchy function $f(z)$, regular in ${\cal D^+}$ and on $C$, has a singularity distribution in ${\cal D}^-$ which can be determined in analytical form in terms of the values $f(t)$ numerically prescribed on $C$, which is Wu's conjecture[1]. It is resolved here for $f(z)$ having (i) a single, (ii) double, or (iii) multiple singularities of the types (I) $\Sigma_j^N M_j(z_j-z)^{k_j}$, (II) $M_\ell\log(z_2-z)$, by having their power series expanded in $z$ and matched on a unit circle $(t=e^{i\theta}, -\pi\leq\theta<\pi$ for contour $C$) with the numerically prescribed Fourier series $f(z)=\Sigma_0^\infty c_ne^{in\theta}$ for solution. The mathematical methods used include (a) complex algebra for cases (i)-(ii), (b) for case (iii) a general asymptotic method developed here for resolution to the Conjecture by induction, and (c) the generalized Hilbert transforms to expound essential singularities. This Conjecture has an advanced version for $f(z)$ to be given only one of its two conjugate functions on $C$ to suffice, and another for the complement function $F(z)$ defined as being regular in domain ${\cal D}^-$ and having singularities in ${\cal D^+}$. These new methods are applicable to all relevant problems in mathematics, engineering and mathematical physics requiring breakthrough by having the exterior singularities resolved.