Irregular singular point - Encyclopedia of Mathematics

There are two non-equivalent definitions of an irregular singular point. According to the first one, t0 is called an irregular singular ...   Login www.springer.com TheEuropeanMathematicalSociety Navigation Mainpage PagesA-Z StatProbCollection Recentchanges Currentevents Randompage Help Projecttalk Requestaccount Tools Whatlinkshere Relatedchanges Specialpages Printableversion Permanentlink Pageinformation Namespaces Page Discussion Variants Views View Viewsource History Actions Irregularsingularpoint FromEncyclopediaofMathematics Jumpto:navigation, search Aconceptthataroseintheanalytictheoryoflinearordinarydifferentialequations.Let$A(t)$bean$(n\timesn)$-matrixthatisholomorphicinapuncturedneighbourhoodof$t_0\neq\infty$andthathasasingularityat$t_0$. Thepoint$t_0$isthencalledasingularpointofthesystem $$\dotx=A(t)x.\label{*}\tag{*}$$ Therearetwonon-equivalentdefinitionsofanirregularsingularpoint.Accordingtothefirstone,$t_0$iscalledanirregularsingularpointof\eqref{*}if$A(t)$hasapoleofordergreaterthanoneat$t_0$(cf.Analytictheoryofdifferentialequations,aswellas[2]).Accordingtotheseconddefinition,$t_0$iscalledanirregularsingularpointof\eqref{*}ifthereisnonumber$\sigma>0$suchthateverysolution$x(t)$growsnotfasterthan$|t-t_0|^{-\sigma}$as$t\tot_0$alongrays(cf.[3]).Thecase$t_0=\infty$canbereducedtothecase$t_0=0$bythetransformation$t\tot^{-1}$.Anirregularsingularpointissometimescalledastrongly-singularpoint(cf.,e.g.,Besselequation).Inaneighbourhoodofanirregularsingularpointthesolutionsadmitasymptoticexpansions;thesewerestudiedbyH.Poincaréforthefirsttime[1]. References [1]H.Poincaré,"Surlesintégralesirrégulièresdeséquationslinéaires"ActaMath.,8(1886)pp.295–344[2]W.Wasov,"Asymptoticexpansionsforordinarydifferentialequations",Interscience(1965)[3]E.A.Coddington,N.Levinson,"Theoryofordinarydifferentialequations",McGraw-Hill(1955)pp.Chapts.13–17 HowtoCiteThisEntry:Irregularsingularpoint.EncyclopediaofMathematics.URL:http://encyclopediaofmath.org/index.php?title=Irregular_singular_point&oldid=44766ThisarticlewasadaptedfromanoriginalarticlebyYu.S.Il'yashenko(originator),whichappearedinEncyclopediaofMathematics-ISBN1402006098.Seeoriginalarticle Retrievedfrom"https://encyclopediaofmath.org/index.php?title=Irregular_singular_point&oldid=44766" Category:TeXdone Thispagewaslasteditedon14February2020,at17:34. Privacypolicy AboutEncyclopediaofMathematics Disclaimers Copyrights Impressum-Legal ManageCookies