Entire function - Knowino

In mathematical analysis and, in particular, in the theory of functions of complex variable, an entire function is a function that is ... Entirefunction FromKnowino Thisisthestableversion,checkedon2July2011. Jumpto:navigation, search Inmathematicalanalysisand,inparticular,inthetheoryoffunctionsofcomplexvariable, anentirefunctionisafunctionthatisholomorphicinthewholecomplexplane [1][2]. Contents 1Examples 1.1Entirefunctions 1.2Non-entirefunctions 2Properties 2.1Powerseries 2.2Unboundedness 2.3Orderofanentirefunction 2.4Rangeofvalues 2.5Cauchyintegral 3References [edit]Examples [edit]Entirefunctions Examplesofentirefunctionsarepolynomialandexponentialfunctions. Allsums,andproductsofentirefunctionsareentire,sothattheentirefunctionsformaC-algebra.Further,compositionsofentirefunctionsarealsoentire. Allthederivativesandsomeoftheintegralsofentirefunctions,forexampletheerrorfunctionerf,sineintegralSiandtheBesselfunctionJ0arealsoentirefunctions. [edit]Non-entirefunctions Ingeneral,neitherseriesnorlimitofasequenceofentirefunctionsneedbeanentirefunction. Theinverseofanentirefunctionneednotbeentire.Usually,inverseofanonlinearfunctionisnotentire.(Theinverseofalinearfunctionisentire).Inparticular,inversesoftrigonometricfunctionsarenotentire. Morenon-entirefunctions:rationalfunctionatanycomplex , , , squareroot,logarithm,functionGamma,tetration. Inparticular,non-analyticfunctionsalsoshouldbequalifiedasnon-entire: , , complexconjugation, modulus, argument, Dirichletfunction. [edit]Properties Theentirefunctionshaveallgeneralpropertiesofotheranalyticfunctions,buttheinfiniterangeofanalyticity enhancesthesetoftheproperties,makingtheentirefunctionsespeciallybeautifulandattractiveforapplications. [edit]Powerseries Theradiusofconvergenceofapowerseriesisthedistancetothenearestsingularity.Therefore,itisinfiniteforentirefunctions. AnyentirefunctioncanbeexpandedineverypointtotheTaylorserieswhichconvergeseverywhere. Thisdoesnotmeanthatonecanalwaysusethepowerseriesforpreciseevaluationofanentirefunction, buthelpsalottoprovethetheorems. [edit]Unboundedness Liouville'stheoremstates:anentirefunctionwhichisboundedmustbeconstant[1]. [edit]Orderofanentirefunction Asallentirefunctions(excepttheconstants)areunbounded,theygrowastheargumentbecomelarge,andcanbecharacterisedbytheirgrowthrate,whichiscalledorder. Letbeentirefunction.Positivenumber iscalledorderoffunction ,ifforallpositivenumbers ,largerthan ,thereexistpositivenumber suchthatforallcomplex suchthat ,therelation holds [3]. Inparticular,allpolynomialshaveorder0;theexponentialhasorder1;anderf,astheGaussianexponential,hasorder2. [edit]Rangeofvalues Picard'slittletheoremstates:anon-constantentirefunctiontakesoneverycomplexnumberasvalue,exceptpossiblyone[2]. Forexample,theexponentialnevertakesonthevalue0. [edit]Cauchyintegral Entirefunction,atanycomplexandanycontourCenclosingthepointzjustonce,canbeexpressedviatheCauchy'stheorem [edit]References ↑1.01.1Conway,JohnB.(1978).FunctionsofOneComplexVariableI,2ndedition.Springer.ISBN0-387-90328-3.  ↑2.02.1Boas,RalphP.(1954).EntireFunctions.AcademicPress.OCLC847696.  ↑Krantz(1999).HandbookofComplexVariables.Boston,MA:Birkhäuser.ISBN0-8176-4011-8.  SomecontentonthispagemaypreviouslyhaveappearedonCitizendium. Retrievedfrom"http://knowino.org/wiki/Entire_function" Category:Mathematics Whatdoyouthinkofthispage? Pleasetakeamomenttoratethispagebelow.Foundaproblem? Presentation: Excellent High Fair Low Poor (unsure) Helpfulness: Excellent High Fair Low Poor (unsure) Personaltools Login/createaccount Namespaces Article Discussion Bibliography Links Variants Views Read Edit Viewhistory Actions Search Navigation Welcome! AboutKnowino Recentchanges Randomarticle Help Community VillageInn Guidelines Toolbox Whatlinkshere Relatedchanges Specialpages Printableversion Permanentlink CitethispagePagerating Thispagewaslastmodifiedon2July2011,at17:11. Thispagehasbeenaccessed14times. ContentisavailableunderCreativeCommonsAttribution–ShareAlike3.0Unported. Privacypolicy AboutKnowino Disclaimers