On the generalized Cauchy function and new Conjecture on its exterior singularities
On the generalized Cauchy function and new Conjecture on its exterior singularities作者机构:California Institute of Technology Pasadena CA 91125 USA
出 版 物:《Acta Mechanica Sinica》 (力学学报(英文版))
年 卷 期:2011年第27卷第2期
页 面:135-151页
核心收录:
学科分类:07[理学] 070104[理学-应用数学] 0802[工学-机械工程] 0701[理学-数学] 0801[工学-力学(可授工学、理学学位)] 0702[理学-物理学]
主 题:Uniform continuity of Cauchy’s function · Uni form convergence of Cauchy’s integral formula · Generalized Hilbert type integral transforms · Functional properties and singularity distributions
摘 要:This article studies on Cauchy’s function f (z) and its integral, (2πi)J[ f (z)] ≡ ■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 D + = [D + + C]. (2) Cauchy’s integral formulas and their derivatives z ∈ D + (or z ∈ D ) are proved to converge uniformly in D + (or in D = [D +C]), 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 general- ized 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 typ- ical 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 sig- nificances of these formulas are illustrated with applicationsto 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 ∈ D + = [D + + C], is presented for resolution as a conjecture.