On uniform continuity of Cauchy's function and uniform convergence of Cauchy's integral formula with applications

TL;DR

This paper investigates the uniform continuity and convergence properties of Cauchy's function and integral formula over the entire complex plane, extending classical results and exploring singularities and applications in special domains.

Contribution

It establishes new uniform convergence theorems for Cauchy's integral and its derivatives under weaker smoothness assumptions, and applies these to generalized Hilbert transforms and singularity analysis.

Findings

Proved uniform continuity of $f(z)$ and its derivatives in the closed domain.

Established uniform convergence of Cauchy's integral and derivatives over the entire plane.

Extended results to special domains and formulated an inverse problem for singularity determination.

Abstract

This study is on Cauchy's function $f(z)$ and its integral, $J[f(z)]\equiv (2\pi i)^{-1}\oint_C f(t)dt/(t-z)$ taken along a closed simple contour $C$, in regard to their comprehensive properties over the entire $z=x+iy$ plane consisted of the open domain ${\cal D}^+$ bounded by $C$ and the open domain ${\cal D}^-$ outside $C$. (i) With $f(z)$ assumed to be $C^n$ ($n$ times continuously differentiable) $\forall z\in {\cal D}^+$ and in a neighborhood of $C$, $f(z)$ and its derivatives $f^{(n)}(z)$ are proved uniformly continuous in the closed domain $\bar{{\cal D}^+}=[\cal D^++C]$. (ii) Under this new assumption, integral $J[f(z)]$ and its derivatives $J_n[f(z)]=d^n J[f(z)]/dz^n$ are proved to converge uniformly in $\bar{{\cal D}^+}$, thereby rendering the integral formula valid over the entire $z$-plane. (iii) The same claims (as for $f(z)$ and $J[f(z)]$) are shown extended to hold for the complement function $F(z)$, defined to be $C^n \forall z\in \bar{{\cal D}^-}=[\cal D^-+C]$, in $\bar{{\cal D}^-}$. (iv) Further, the singularity distribution of $f(z)$ in ${\cal D}^-$ (existing unless $f(z)\equiv$ const.in the $z$-plane) is elucidated by considering the direct problem exemplified with several typical singularities prescribed in ${\cal D}^-$. (v) The uniform convergence theorems for $f(z)$ and $F(z)$ shown for contour $C$ of arbitrary shape are adapted to apply to special domains in the upper or lower half $z$-planes and those inside and outside the unit circle $|z|=1$ to achieve the generalized Hilbert transforms for these cases. (vi) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function $f(z)$ in domain ${\cal D}^-$ is presented for resolution as a conjecture.