On resolution to Wu's Conjecture

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.