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

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.