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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 <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" />.
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Painlevé's theorem on the solutions to analytic differential equations. The solutions to the differential equation $P(w',w,z)=0$ 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 $P$ is a polynomial in the unknown function $w$ and its derivative $w'$, while $P$ is an analytic function in the independent variable $z$.
  
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]]].
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Painlevé's theorem on analytic continuation. If $\Gamma$ is a rectifiable Jordan curve lying in a domain $D$ in the complex $z$-plane and if a function $f(z)$ is continuous in $D$ and analytic in $D\setminus\Gamma$, then $f(z)$ is an [[Analytic function|analytic function]] in the entire domain $D$ [[#References|[1]]], [[#References|[2]]].
  
 
====References====
 
====References====
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For Painlevé's theorem on differential equations see also [[#References|[a1]]], [[#References|[a4]]].
 
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]]].
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If in 2) $\Gamma$ is not required to be rectifiable, the analytic continuation need not be possible, cf. [[#References|[a1]]], [[#References|[a2]]].
  
 
====References====
 
====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>
 
<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>

Latest revision as of 14:46, 17 July 2014

Painlevé's theorem on the solutions to analytic differential equations. The solutions to the differential equation $P(w',w,z)=0$ cannot have movable (i.e. dependent on an arbitrary constant) essentially-singular points (cf. Movable singular point) and transcendental branch points, where $P$ is a polynomial in the unknown function $w$ and its derivative $w'$, while $P$ is an analytic function in the independent variable $z$.

Painlevé's theorem on analytic continuation. If $\Gamma$ is a rectifiable Jordan curve lying in a domain $D$ in the complex $z$-plane and if a function $f(z)$ is continuous in $D$ and analytic in $D\setminus\Gamma$, then $f(z)$ is an analytic function in the entire domain $D$ [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) $\Gamma$ 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)
How to Cite This Entry:
Painlevé theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Painlev%C3%A9_theorem&oldid=32468
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article