Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T13:19:30.207Z Has data issue: false hasContentIssue false

On the Location of Singularities of a Class of Elliptic Partial Differential Equations in Four Variables

Published online by Cambridge University Press:  20 November 2018

R. P. Gilbert*
Affiliation:
Georgetown University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we shall investigate the singular behaviour of the solutions to the elliptic equation

(1.1)

where A (r2), C(r2) are entire functions of the complex variable

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Behnke, H. and Grauert, H., Analysis in non-compact complex spaces (in Analytic functions), No. 24 (Princeton, 1960), pp. 1144.Google Scholar
2. Behnke, H. and Thullen, P., Théorie der Funktionen mehrerer komplexer Verànderlichen (Berlin, 1934).Google Scholar
3. Bergman, S., Integral operators in the theory of linear partial differential equations (Berlin, 1960).Google Scholar
4. Bergman, S., Zur Théorie der algebraischen Potential FunktiGnen des dreidimensional Raumes, Math. Ann., 99 (1929), 629659, and 101 (1929), 534538.Google Scholar
5. Bergman, S., Multivalued harmonic functions in three variables, Comm. Pure Appl. Math., 9 (1956), 327338.Google Scholar
6. Bergman, S., Operators generating solutions of certain differential equations in three variables and their properties, Scripta Math., 26 (1961), 531.Google Scholar
7. Bergman, S., Some properties of a harmonic function of three variables given by its series development, Arch. Rational Mech. Anal., 8 (1961), 207222.Google Scholar
8. Bergman, S., Integral operators in the study of an algebra and a coefficient problem in the theory of three dimensional harmonic functions, Duke Math. J., 30 (1963), 447460.Google Scholar
9. Bergman, S., On the coefficient problem on the theory of a system of linear partial differential equations, J. Analyse Math., 11 (1963), 249274.Google Scholar
10. Bochner, S. and Martin, W. T., Several complex variables (Princeton, 1948).Google Scholar
11. Carathéodory, C., Function theory, Vols I, II (New York, 1950).Google Scholar
12. Erd, A.élyi, Magnus, W., Oberhettinger, F., and Tricomi, F. G., Higher transcendental functions II (New York, 1953).Google Scholar
13. Gilbert, R. P., Singularities of three-dimensional harmonic functions, Pacific J. Math., 10 (1960), 12431255.Google Scholar
14. Gilbert, R. P., Singularities of solutions to the wave equation in three dimensions, J. Reine Angew. Math., 205 (1960), 7581.Google Scholar
15. Gilbert, R. P., On harmonic functions of four variables with rational P4-associates, Pacific J. Math., 13 (1963), 7996.Google Scholar
16. Gilbert, R. P., Harmonic functions in four variables with algebraic and rational Pi-associates, Ann. Polon. Math., 15 (1964), 273287.Google Scholar
17. Gilbert, R. P., Multivalued harmonic functions in four variables, to appear in J. Analyse Math.Google Scholar
18. Gilbert, R. P., On a class of elliptic partial differential equations in four variables, Pacific J. Math., 14 (1964), 12231236.Google Scholar
19. Gilbert, R. P. and Howard, H. C., On a class of elliptic partial differential equations, Technical Note BN-344, December 1963.Google Scholar
20. Gilbert, R. P. and Howard, H. C., Integral operator methods for generalized axially symmetric potentials in (n + 1)- variables, to appear in J. Austral. Math. Soc.Google Scholar
21. Hadamard, J., Essai sur Vétude des fonctions données par leurs développements de Taylor, J. Math., 4, 8 (1892), 101186.Google Scholar
22. Kreyszig, E., Coefficient problems in systems of partial differential equations, Arch. Rational Mech. Anal, 1 (1958), 283294.Google Scholar
23. Kreyszig, E., On singularities of solutions of partial differential equations in three variables, Arch. Rational Mech. Anal, 2 (1958), 151159.Google Scholar
24. Kreyszig, E., Kanonische integral Operatoren zur Erzeugung harmonischer Funktionen von vier Verànderlichen, Arch. Math., 14 (1963), 193203.Google Scholar
25. Mandelbrojt, S., Théorème général fournissant Vargument des points singuliers situés sur le cercle de convergence d'une série de Taylor, C. R. Acad. Sci. Paris, 204 (1937), 14561458.Google Scholar
26. de Mises, R., La base géométrique du théorème de M. Mandelbrojt sur les points singuliers d'une fonction analytique, C.R. Acad. Sci. Paris, 205 (1938), 13531355.Google Scholar
27. Mitchell, J., Integral theorems for harmonic vectors in three real variables, Math. Z., 82 (1963), 314334.Google Scholar
28. Mitchell, J., Representation theorems for solutions of linear partial differential equations in three variables, Arch. Rational Mech. Anal, 3 (1959), 439459.Google Scholar
29. Oka, K., Sur les fonctions analytiques de plusieurs variables (Tokyo, 1960).Google Scholar
30. Osgood, W. F., Lehrbuch der Funktionentheorie, Vol. 2 (2nd éd., Leipzig, 1929).Google Scholar
31. Poincar, H.é, Sur les fonctions de deux variables, Acta Math., 2 (1883), 6472.Google Scholar
32. White, A., Singularities of harmonie functions of three real variables generated by Whittaker- Bergman operators, Ann. Polon. Math., 10 (1961), 82100.Google Scholar