View source for Painlevé theorem

Painlevé's theorem on the solutions to analytic differential equations. The solutions to the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p0711001.png" /> cannot have movable (i.e. dependent on an arbitrary constant) essentially-singular points (cf. [[Movable singular point|Movable singular point]]) and transcendental branch points, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p0711002.png" /> is a polynomial in the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p0711003.png" /> and its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p0711004.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p0711005.png" /> is an analytic function in the independent variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p0711006.png" />.

Painlevé's theorem on analytic continuation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p0711007.png" /> is a rectifiable Jordan curve lying in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p0711008.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p0711009.png" />-plane and if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p07110010.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p07110011.png" /> and analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p07110012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p07110013.png" /> is an [[Analytic function|analytic function]] in the entire domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p07110014.png" /> [[#References|[1]]], [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Painlevé,   "Sur les lignes singulières des fonctions analytiques" , Paris  (1887)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Painlevé,   "Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (1895)" , Paris  (1897)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Golubev,   "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR></table>



====Comments====
For Painlevé's theorem on differential equations see also [[#References|[a1]]], [[#References|[a4]]].

If in 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071100/p07110015.png" /> is not required to be rectifiable, the analytic continuation need not be possible, cf. [[#References|[a1]]], [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Garnett,   "Analytic capacity and measure" , ''Lect. notes in math.'' , '''297''' , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Wermer,   "Banach algebras and several complex variables" , Springer  (1976)  pp. Chapt. 13</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)  pp. §§3.6, 3.51, 4.7, A.5</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Hille,   "Lectures on ordinary differential equations" , Addison-Wesley  (1969)</TD></TR></table>