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On the generalized Cauchy function and new Conjecture on its exterior singularities | |
Wu Theodore Yaotsu; Wu, TY (reprint author), CALTECH, Pasadena, CA 91125 USA | |
Source Publication | Acta Mechanica Sinica |
2011 | |
Volume | 27Issue:2Pages:135-151 |
ISSN | 0567-7718 |
Abstract | This article studies on Cauchy's function f(z) and its integral, (2 pi i)J[f(z)] equivalent to closed integral(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 < infinity-times continuously differentiable) for all z is an element of D(+) and in a neighborhood of C, f (z) and its derivatives f((n))(z) are proved uniformly continuous in the closed domain <(D(+))over bar> = [D(+) + C]. (2) Cauchy's integral formulas and their derivatives for all z is an element of D(+) (or for all z is an element of D(-)) are proved to converge uniformly in (D(+)) over bar (or in (D(-)) over bar = [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)for all z is an element of 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)for all z is an element of (D(+)) over bar = [D(+) + C], is presented for resolution as a conjecture. |
Keyword | Uniform Continuity Of Cauchy's Function Uniform Convergence Of Cauchy's Integral Formula Generalized Hilbert-type Integral Transforms Functional Properties And Singularity Distributions Solitary Waves |
Subject Area | Engineering ; Mechanics |
DOI | 10.1007/s10409-011-0446-8 |
URL | 查看原文 |
Indexed By | SCI ; EI |
Language | 英语 |
WOS ID | WOS:000292036300001 |
WOS Keyword | SOLITARY WAVES |
WOS Research Area | Engineering ; Mechanics |
WOS Subject | Engineering, Mechanical ; Mechanics |
Classification | 二类 |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://dspace.imech.ac.cn/handle/311007/44992 |
Collection | 环境力学重点实验室(2009-2011) |
Corresponding Author | Wu, TY (reprint author), CALTECH, Pasadena, CA 91125 USA |
Recommended Citation GB/T 7714 | Wu Theodore Yaotsu,Wu, TY . On the generalized Cauchy function and new Conjecture on its exterior singularities[J]. Acta Mechanica Sinica,2011,27(2):135-151. |
APA | Wu Theodore Yaotsu,&Wu, TY .(2011).On the generalized Cauchy function and new Conjecture on its exterior singularities.Acta Mechanica Sinica,27(2),135-151. |
MLA | Wu Theodore Yaotsu,et al."On the generalized Cauchy function and new Conjecture on its exterior singularities".Acta Mechanica Sinica 27.2(2011):135-151. |
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