Sets of Minimum Capacity, Padé Approximants and the Bubble Problem
We give a characterization of a set of minimum capacity whose components include certain prescribed points, but the proof of uniqueness of such a set remains to be completed. It is speculated that Padé approximants to certain functions with branch points converge away from a corresponding set of minimum capacity. In some cases this speculation has been proved. These ideas are used to study the expansion for a function describing the shape of a cylindrical bubble rising through a fluid.
Unable to display preview. Download preview PDF.
- 1.Nuttall, J.: “On sets of minimum capacity” (unpublished)Google Scholar
- 2.Hille, E.: “Analytic Function Theory”, Vol. 2, Ginn and Co.,Waltham, Mass. 1962, p 275Google Scholar
- 3.Schiffer, M.: appendix to Courant, R: “Dirichlet’s Principle,Conformal Mapping and Minimal Surfaces,” Interscience Publishers Inc., New York, 1950.Google Scholar
- 4.Bergman, S.: “The Kernel Function and Conformal Mapping”,American Mathematical Society, Providence, 1970.Google Scholar
- 5.Siegel,C.L.: “Topics in Complex Function Theory,” Vol. 2, Interscience Publishers Inc., New York, 1970.Google Scholar
- 7.Szego, G.: “Orthogonal Polynomials”, American Mathematical Society, New York, 1959.Google Scholar
- 8.Dumas, S.: “Sur le Developpement des Fonctions Elliptiques en Fractions Continues,” Thesis, Zurich, 1908Google Scholar
- 9.Walters, J.K., and Davidson, J.F..: J, Fluid Mech, 12, pp 408–416Google Scholar
- 10.Pommerenke, C.: “Univalent Functions,” Vandenhoeck and Ruprecht, Göttingen 1975Google Scholar
- 11.Freeman, D.F., and Nuttall, J.: “Time Development of a cylindrical fluid without viscosity”, (unpublished)Google Scholar
- 12.Gammel, J.L.: in “Pade Approximants and their Applications” edited by P.R. Graves-Morris, Academic Press, London, 1973, pp 3–9Google Scholar
- 13.Hunter, D.L., and Baker, G.A.: “Methods of Series Analysis, III: Integral Approximant Methods”, preprint, 1978Google Scholar