On resolution to Wu’s conjecture on Cauchy function’s exterior singularities

Abstract

This is a series of studies on Wu’s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy’s function f(z) (z = x + iy) and its integral $J\left[ {f\left( z \right)} \right] \equiv \left( {2\pi i} \right)^{ - 1} \oint_C {f\left( t \right)\left( {t - z} \right)^{ - 1} dt}$ taken along the unit circle as contour C, inside which (the open domain D ⁺) f(z) is regular but has singularities distributed in open domain D ⁻ outside C. Resolution is given to the inverse problem that the singularities of f(z) can be determined in analytical form in terms of the values f(t) of f(z) numerically prescribed on C (|t| = 1), as so enunciated by Wu’s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle, for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction, and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics, engineering and technology in analogy with resolving the inverse problem presented here.