Entire Function -- from Wolfram MathWorld

If a complex function is analytic at all finite points of the complex plane C, then it is said to be entire, sometimes also called "integral" (Knopp 1996, ... Algebra AppliedMathematics CalculusandAnalysis DiscreteMathematics FoundationsofMathematics Geometry HistoryandTerminology NumberTheory ProbabilityandStatistics RecreationalMathematics Topology AlphabeticalIndex NewinMathWorld CalculusandAnalysis ComplexAnalysis GeneralComplexAnalysis CalculusandAnalysis ComplexAnalysis EntireFunctions MathWorldContributors Bhatt MathWorldContributors Renze More...Less... EntireFunction Download Wolfram Notebook Ifacomplexfunctionisanalyticatallfinitepointsofthecomplexplane,thenitissaid tobeentire,sometimesalsocalled"integral"(Knopp1996,p. 112). Anypolynomial isentire. Examplesofspecificentirefunctionsaregiveninthefollowingtable. functionsymbolAiryfunctions,Airyfunctionderivatives,AngerfunctionBarnesG-functionbeiberBesselfunction ofthefirstkindBesselfunction ofthesecondkindBeurling'sfunctioncosinecoversineDawson'sintegralerferfcerfiexponentialfunctionFresnelintegrals,gammafunctionreciprocalgeneralizedhypergeometricfunctionhaversinehyperboliccosinehyperbolicsineJacobiellipticfunctions,,,,,,,,,,,JacobithetafunctionsJacobithetafunctionderivativesMittag-LefflerfunctionmodifiedStruvefunctionNevillethetafunctions,,,Shisinesine integralspherical BesselfunctionofthefirstkindStruvefunctionversineWeberfunctionsWrightfunctionxi-function Liouville'sboundednesstheoremstatesthataboundedentirefunctionmustbeaconstant function. SeealsoAnalyticFunction,FiniteOrder,HadamardFactorizationTheorem, HolomorphicFunction,Liouville's BoundednessTheorem,MeromorphicFunction, WeierstrassProductTheorem ExplorewithWolfram|Alpha Morethingstotry: {{2,-1,1},{0,-2,1},{1,-2,0}}.{x,y,z} fcclatticepointgroups grad(F.G) ReferencesKnopp,K."EntireTranscendentalFunctions."Ch. 9inTheory ofFunctionsPartsIandII,TwoVolumesBoundasOne,PartI.NewYork: Dover,pp. 112-116,1996.Krantz,S. G."EntireFunctions andLiouville'sTheorem."§3.1.3inHandbook ofComplexVariables.Boston,MA:Birkhäuser,pp. 31-32,1999.Referenced onWolfram|AlphaEntireFunction Citethisas: Weisstein,EricW."EntireFunction." FromMathWorld--AWolframWebResource.https://mathworld.wolfram.com/EntireFunction.html Subjectclassifications CalculusandAnalysis ComplexAnalysis GeneralComplexAnalysis CalculusandAnalysis ComplexAnalysis EntireFunctions MathWorldContributors Bhatt MathWorldContributors Renze More...Less... Created,developedandnurturedbyEricWeissteinatWolframResearch