Difference between revisions of "Painlevé theorem"
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Painlevé's theorem on the solutions to analytic differential equations. The solutions to the differential equation cannot have movable (i.e. dependent on an arbitrary constant) essentially-singular points (cf. Movable singular point) and transcendental branch points, where
is a polynomial in the unknown function
and its derivative
, while
is an analytic function in the independent variable
.
Painlevé's theorem on analytic continuation. If is a rectifiable Jordan curve lying in a domain
in the complex
-plane and if a function
is continuous in
and analytic in
, then
is an analytic function in the entire domain
[1], [2].
References
[1] | P. Painlevé, "Sur les lignes singulières des fonctions analytiques" , Paris (1887) |
[2] | P. Painlevé, "Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (1895)" , Paris (1897) |
[3] | V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) |
Comments
For Painlevé's theorem on differential equations see also [a1], [a4].
If in 2) is not required to be rectifiable, the analytic continuation need not be possible, cf. [a1], [a2].
References
[a1] | J.B. Garnett, "Analytic capacity and measure" , Lect. notes in math. , 297 , Springer (1972) |
[a2] | J. Wermer, "Banach algebras and several complex variables" , Springer (1976) pp. Chapt. 13 |
[a3] | E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) pp. §§3.6, 3.51, 4.7, A.5 |
[a4] | E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1969) |
Painlevé theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Painlev%C3%A9_theorem&oldid=22879