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, $J[f(z)]\equiv (2\pi i)^{-1}\oint_C f(t)dt/(t-z)$takenalongaclosedsimplecontour$C$, in regard to their comprehensive properties over the entire $z=x+iy$ plane consisted of the open domain ${\mathcal{D}}^{+}$ bounded by $C$ and the open domain ${\mathcal{D}}^{-}$ outside $C$. (i) With $f(z)$ assumed to be $C^n$ ($n$ times continuously differentiable) $\mathrm{\forall }z\in {\mathcal{D}}^{+}$ and in a neighborhood of $C$, $f(z)$ and its derivatives $f^{(n)}(z)$ are proved uniformly continuous in the closed domain $\overline{{\mathcal{D}}^{+}}=[{\mathcal{D}}^{+}+\mathcal{C}]$. (ii) Under this new assumption, integral $J[f(z)]$ and its derivatives $J_n[f(z)]=d^{n}J[f(z)]\mathrm{/}d{z}^n$ are proved to converge uniformly in $\overline{{\mathcal{D}}^{+}}$, 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 $C^n\mathrm{\forall }z\in \overline{{\mathcal{D}}^{-}}=[{\mathcal{D}}^{-}+\mathcal{C}]$, in $\overline{{\mathcal{D}}^{-}}$. (iv) Further, the singularity distribution of $f(z)$ in ${\mathcal{D}}^{-}$ (existing unless $f(z)\equiv$ const.in the $z$-plane) is elucidated by considering the direct problem exemplified with several typical singularities prescribed in ${\mathcal{D}}^{-}$. (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 $\mathrm{\mid }z\mathrm{\mid }=1$ 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.