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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 PublicationActa Mechanica Sinica
2011
Volume27Issue:2Pages:135-151
ISSN0567-7718
AbstractThis 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.
KeywordUniform 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 AreaEngineering ; Mechanics
DOI10.1007/s10409-011-0446-8
URL查看原文
Indexed BySCI ; EI
Language英语
WOS IDWOS:000292036300001
WOS KeywordSOLITARY WAVES
WOS Research AreaEngineering ; Mechanics
WOS SubjectEngineering, Mechanical ; Mechanics
Classification二类
Citation statistics
Cited Times:1[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Identifierhttp://dspace.imech.ac.cn/handle/311007/44992
Collection环境力学重点实验室(2009-2011)
Corresponding AuthorWu, 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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