The definition of "entire function" - Mathematics Stack Exchange

I am reading about the definition of "entire functions" : "If a complex function is analytic at all finite points of the complex plane C, ... MathematicsStackExchangeisaquestionandanswersiteforpeoplestudyingmathatanylevelandprofessionalsinrelatedfields.Itonlytakesaminutetosignup. Signuptojointhiscommunity Anybodycanaskaquestion Anybodycananswer Thebestanswersarevotedupandrisetothetop Home Public Questions Tags Users Unanswered Teams StackOverflowforTeams –Startcollaboratingandsharingorganizationalknowledge. CreateafreeTeam WhyTeams? Teams CreatefreeTeam Teams Q&Aforwork Connectandshareknowledgewithinasinglelocationthatisstructuredandeasytosearch. LearnmoreaboutTeams Thedefinitionof"entirefunction" AskQuestion Asked 5yearsago Modified 1year,4monthsago Viewed 12ktimes 10 4 $\begingroup$ Iamreadingaboutthedefinitionof"entirefunctions": "Ifacomplexfunctionisanalyticatallfinitepointsofthecomplexplane$\mathbb{C}$,thenitissaidtobeentire..." Infact,I'dliketounderstandthisdefinition.ThusIwishahelptorespondmyquestions. Areallanalyticfunctionson$\mathbb{C}$entire? Whydoweneedtousethisdefinition? Thankyouverymuchforallofyouranswers! complex-analysisfunctional-analysisanalytic-number-theory Share Cite Follow editedApr16,2021at18:05 BillDubuque 259k3737goldbadges266266silverbadges869869bronzebadges askedAug21,2017at13:02 ToThichToanToThichToan 63311goldbadge44silverbadges1313bronzebadges $\endgroup$ 5 3 $\begingroup$ Thequestioninpoint1doesn'tmakesense.Sincethisisaquestionaboutunderstanding,youshouldprobablyclarifywhatyoumeanratherthanleaveittopeopletoguess.(manywon'tevenrealizethey'reguessing!)Unless,Isuppose,thequestionyouneedanswerediswhat'swrongwithyourgrammar(butifsoyoushouldclarify!). $\endgroup$ – user14972 Aug21,2017at13:07 3 $\begingroup$ Entirefunctionsareverynice-theycanallbewrittenasapowerseriesthatconvergeseverywhereon$\mathbbC$.Theyextendtheringofpolynomialsinusefulways. $\endgroup$ – ThomasAndrews Aug21,2017at13:12 4 $\begingroup$ Becarefulwithprepositions.Wesayafunctionisanalyticatapoint,orwesayafunctionisanalyticonaset(ifitisanalyticateverypointintheset.)Soanentirefunctionisanalyticon$\mathbbC$,notat$\mathbbC$. $\endgroup$ – ThomasAndrews Aug21,2017at13:13 1 $\begingroup$ Thankyou@ThomasAndrews!Irepairedthatpoint! $\endgroup$ – ToThichToan Aug22,2017at0:25 $\begingroup$ @Hurkyl:Thankyoufortheadvice.Iwillwritecarefullynexttime! $\endgroup$ – ToThichToan Aug22,2017at0:27 Addacomment  |  3Answers 3 Sortedby: Resettodefault Highestscore(default) Datemodified(newestfirst) Datecreated(oldestfirst) 15 $\begingroup$ Thereare3definitionsofentirefunctions,allequivalent: $f(z)=\sum_{n=0}^\inftya_nz^n$isentireiffitconvergesforevery$z\in\mathbb{C}$(seetheradiusofconvergenceofpowerseries) $f$isentireiffitiseverywhereanalytic,thatisforevery$z_0\in\mathbb{C}$thereis$r>0$suchthat$f(z)=\sum_{n=0}^\infty\frac{f^{(n)}(z_0)}{n!}(z-z_0)^n$for$|z-z_0|