Abstract
This is a series of studies on Wu’s conjecture and on its resolution to be presented herein. Both are devoted to expound all the comprehensive properties of Cauchy’s function f(z) (z = x + iy) and its integral \(J\left[ {f\left( z \right)} \right] \equiv \left( {2\pi i} \right)^{ - 1} \oint_C {f\left( t \right)\left( {t - z} \right)^{ - 1} dt}\) taken along the unit circle as contour C, inside which (the open domain D +) f(z) is regular but has singularities distributed in open domain D − outside C. Resolution is given to the inverse problem that the singularities of f(z) can be determined in analytical form in terms of the values f(t) of f(z) numerically prescribed on C (|t| = 1), as so enunciated by Wu’s conjecture. The case of a single singularity is solved using complex algebra and analysis to acquire the solution structure for a standard reference. Multiple singularities are resolved by reducing them to a single one by elimination in principle, for which purpose a general asymptotic method is developed here for resolution to the conjecture by induction, and essential singularities are treated with employing the generalized Hilbert transforms. These new methods are applicable to relevant problems in mathematics, engineering and technology in analogy with resolving the inverse problem presented here.
Similar content being viewed by others
References
Wu, Th.Y.: On the generalized Cauchy function and new conjecture on its exterior singularities. Acta Mech. Sin. 27(2), 135–151 (2011)
Wu, Th.Y.: On resolution to Wu’s conjecture. 2009. arXiv: 0909.0298v1
Stokes, G.G.: On the theory of oscillatory waves. Trans. Cambridge Phil. Soc. 8, 441–455 (1847)
Grant, M.A.: The singularity at the crest of a finite amplitude progressive Stokes wave. J. Fluid Mech. 59(2), 257–262 (1973)
Schwartz, L.M.: Computer extension and analytic continuation of Stokes expansion of gravity waves. J. Fluid Mech. 62(3), 553–578 (1974)
Tanveer, S.: Singularities in water waves and Rayleigh-Taylor instability. Proc. R. Soc. Lond. A 435, 137–158 (1991)
Domb, C., Sykes, M.F.: On the susceptibility of a ferromagnetic above the Curies point. Proc. R. Soc. Lond. A 240, 214–218 (1957)
Li, J.: Singularity criteria for perturbation series. Scientia Sinica A 25(6), 593–600 (1982)
Van Dyke, M.D.: Analysis and improvement of perturbation series. Q. J. Mech. Appl. Math. 27, 423–440 (1974)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wu, T.Y. On resolution to Wu’s conjecture on Cauchy function’s exterior singularities. Acta Mech Sin 27, 309–317 (2011). https://doi.org/10.1007/s10409-011-0465-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-011-0465-5