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

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

2007-10-31

arxiv.org/abs/0710.5790v2

Mathematics

Complex Variables

21 pages, 2 figures

Scientific paper

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.

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