Abstract
The analytic continuation of a solution of the generalized axially symmetric Helmholtz equationu xx + u yy + (2α/x)u x + k 2 u = 0is examined. A representation in terms of boundary data and the complex Riemann function is given for the continuation of the solution to an analytic boundary value problem; this also provides the solution of the analytic Cauchy problem on an analytic arc. Integral representations are found for the Riemann function, and the axial behaviour of the Riemann function is determined and used to recover a representation for the solution in terms of analytic axial data, as given originally by Henrici. For an exterior boundary value problem in which the axial values of the solution are defined on two disjoint, semi-infinite segments of the axis, it is shown that the two functions are not analytic continuations of one an-other and that a certain linear combination of them is an entire function. As an example, for α = 1/2 it is shown that the continuation of an exterior solution for a prolate spheroidal boundary is logarithmically infinite on the interfocal segment. A further special case, one that involves wave scattering by slender bodies of revolution for which the solution may be represented as a superposition over axial singularities, is briefly examined; properties of the axial values which are forced by this representation are determined and, where comparison is possible, shown to be consistent with the present work.
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References
Bateman, H., Partial Differential Equations of Mathematical Physics. New York: Dover Publications, 1944.
Chwang, A. T., & T. Y.-T. Wu, Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech. 67, 787–815 (1975).
Collins, W. D., On the solution of some axisymmetric boundary value problems by means of integral equations. V. Some scalar diffraction problems for circular disks. Quart. J. Mech. Appl. Math. 14, 101–117 (1961).
Colton, D., John's decomposition theorem for generalized metaharmonic functions. J. London Math. Soc. (2) 1, 737–742 (1969).
Copson, E. T., On the Riemann-Green function. Arch. Rational Mech. Anal. 1, 324–348 (1958).
Erdélyi, A., Singularities of generalized axially symmetric potentials. Comm. Pure Appl. Math. 9, 403–414 (1956).
Gakhov, F. D., Boundary Value Problems. London: Pergamon Press, 1966.
Garabedian, P. R., Partial Differential Equations. New York: John Wiley and Sons, Inc., 1964.
Geer, J., Uniform asymptotic solutions for potential flow about a slender body of revolution. J. Fluid Mech. 67, 817–827 (1975).
Geer, J., Stokes flow past a slender body of revolution. J. Fluid Mech. 78, 577–600 (1976).
Geer, J., The scattering of a scalar wave by a slender body of revolution. SIAM J. Appl. Math. 34, 348–370 (1978).
Gilbert, R. P., Function Theoretic Methods in Partial Differential Equations. New York: Academic Press, 1969.
Handelsman, R. A., & J. B. Keller, Axially symmetric potential flow around a slender body. J. Fluid Mech. 28, 131–147 (1967).
Handelsman, R. A., & J. B. Keller, The electrostatic field around a slender conducting body of revolution. SIAM J. Appl. Math. 15, 824–841 (1967).
Heins, A. E., Axially-symmetric boundary-value problems. Bull. Amer. Math. Soc. 71, 787–808 (1965).
Heins, A. E., The fundamental solution for the axially symmetric wave equation. Arch. Rational Mech. Anal. 67, 83–100 (1977).
Heins, A. E., & R. C. MacCamy, Integral representations of axially symmetric potential functions. Arch. Rational Mech. Anal. 13, 371–385 (1963).
Henrici, P., Zur Funktionentheorie der Wellengleichung. Comment. Math. Helv. 27, 235–293 (1953).
Henrici, P., A survey of I. N. Vekua's theory of elliptic partial differential equations with analytic coefficients. Z. Angew. Math. Phys. 8, 169–203 (1957).
Kellogg, O. D., Foundations of Potential Theory. New York: Dover Publications, 1953.
Millar, R. F., Singularities of solutions to exterior analytic boundary value problems for the Helmholtz equation in three independent variables. II: The axisymmetric boundary. SIAM J. Math. Anal. 10, 682–694 (1979).
Millar, R. F., The analytic continuation of solutions to elliptic boundary value problems in two independent variables. J. Math. Anal. Appl. 76, 498–515 (1980).
Vekua, I. N., New Methods for Solving Elliptic Equations. Amsterdam: North-Holland, 1967.
Watson, G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Second Edition, (1944).
Weinacht, R. J., Fundamental solutions for a class of singular equations. Contributions to Differential Equations 3, 43–55 (1964).
Weinstein, A., Discontinuous integrals and generalized potential theory. Trans. Amer. Math. Soc. 63, 342–354 (1948).
Weinstein, A., Generalized axially symmetric potential theory. Bull. Amer. Math. Soc. 59, 20–38 (1953).
Whittaker, E. T., & G. N. Watson, A Course of Modern Analysis. Cambridge University Press, Fourth Edition, (1927).
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Communicated by M. M. Schiffer
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Millar, R.F. The analytic continuation of solutions of the generalized axially symmetric Helmholtz equation. Arch. Rational Mech. Anal. 81, 349–372 (1983). https://doi.org/10.1007/BF00250860
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DOI: https://doi.org/10.1007/BF00250860