On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f (z) and its integral, $ (2pi i)J[f(z)] equiv oint_C {f(t)dt/(t - z)} $ (2pi i)J[f(z)] equiv 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 simply connected open domain D ⁺ bounded by C and the open domain D ⁻ outside C. (1) With f (z) assumed to be C ⁿ (n < ∞-times continuously differentiable) ∀ _z ∈ D ⁺ and in a neighborhood of C, f (z) and its derivatives f ⁽ⁿ⁾(z) are proved uniformly continuous in the closed domain $ overline {D^ + } = [D^ + + C] $ overline {D^ + } = [D^ + + C] . (2) Cauchy’s integral formulas and their derivatives ∀z ∈ D ⁺ (or ∀z ∈ D ⁻) are proved to converge uniformly in $ overline {D^ + } $ overline {D^ + } (or in $ overline {D^ + } $ overline {D^ + } ), respectively, thereby rendering the integral formulas valid over the entire z-plane. (3) 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 ∀z ∈ D ⁻ and about C. (4) The uniform convergence theorems for f (z) and F(z) shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle |z| = 1 such that the four generalized Hilbert-type integral transforms are proved. (5) Further, the singularity distribution of f(z) in D ⁻ is elucidated by considering the direct problem exemplified with several typical singularities prescribed in D ⁻. (6) A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. (7) Physical significances of these formulas are illustrated with applications to nonlinear airfoil theory. (8) Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f(z) in domain D ⁻, based on the continuous numerical value of f(z)∀z ∈ $ overline {D^ + } = [D^ + + C] $ overline {D^ + } = [D^ + + C] , is presented for resolution as a conjecture.