Review Exam I Complex Analysis. Cauchy s Integral Formula (#0). Let G be a region in C, let Bar (, ) G and let γ be the circle C(a,r), oriented.

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1 Revew Exa I Coplex Aalyss Uderled Deftos: May be ased for o exa Uderled Propostos or Theores: Proofs ay be ased for o exa Cauchy s Itegral Forula (#) Let G be a rego C, let Bar (, ) G ad let be the crcle C(a,r), oreted f( w) postvely Let f A ( G) The, for Bar (, ), f dw π Cauchy s Theore (#) Let G be a rego C, let Bar (, ) G ad let be a closed rectfable curve B(a,r) Let f A ( G) The, f Chapter 45 Defto for a closed rectfable curve lyg a rego G f Lea 5 Let be a rectfable curve ad let ϕ be cotuous o { } For each, let ϕ( w) ' f () for The, ad dw C \{} f A ( C \{ }) f f () + Cauchy s Itegral Forula (#) Let G be a rego C ad let f A ( G) Let be a closed rectfable curve f( w) G such that The, for G \{}, ( ; ) f dw π Cauchy s Itegral Forula (#) Let G be a rego C ad let f A ( G) Let be a closed,,, rectfable curves G such that The, for G \ { }, f ( w) ( ; ) f dw π Cauchy s Theore (#) Let G be a rego C ad let f A ( G) Let be a closed rectfable,,, curves G such that The, f Corollary Let G be a rego C ad let f A ( G) Let be a closed rectfable curve G such that ( )! f ( w) The, for G \{}, ( ; ) f dw π + Morera s Theore Let G be a rego C ad let f C ( G) Suppose that f for every tragular path T G The, f A ( G) T

2 Chapter 46 Defto ~ for two closed rectfable paths, lyg a rego G f Proposto ~ s a equvalece relato Exaples of pars of curves (ad the overlyg rego G ) whch are hootopc ad of pars of curves,, (ad the overlyg rego G ) whch are ot hootopc Probles about costructg the hootopy fucto for a par of curves, (ad the overlyg rego G ) whch are hootopc Defto A set S s covex f Defto A set S s starle (wrt a) f Exaples of sets whch are covex / starle ad of sets whch are ot covex / starle Cauchy s Theore (#) Let G be a rego C ad let be a closed rectfable curve G If ~, the Corollary Let G be a rego C whch s starle ad let f A ( G) Let be a closed rectfable curve G The, f Cauchy s Theore (#3) Let G be a rego C ad let f A ( G) Let Let, be closed rectfable curves G such that ~ The, f f Corollary Let G be a rego C ad let f A ( G) Let be a closed rectfable curve G such that ~ The, f Defto for two rectfable paths lyg a rego G f, ~ FEP Exaples of pars of curves, (ad the overlyg rego G ) whch are FEP-hootopc ad of pars of curves, (ad the overlyg rego G ) whch are ot FEP-hootopc Idepedece of Path Theore Let G be a rego C ad let f A ( G) Let Let, be rectfable curves G such that The, f f ~ FEP

3 Defto A rego G s sply coected f Exaples of regos whch are sply coected ad of regos whch are ot sply coected Cauchy s Theore (#4) Let G be a rego C whch s sply coected ad let f A ( G) Let be a closed rectfable curve G The, f Corollary Let G be a rego C whch s sply coected ad let f A ( G) The, f has a prtve o G Defto A fucto f has a brach-of-log f o a rego G f Corollary Let G be a rego C whch s sply coected ad let f A ( G) If f o G, the f has a brach-of-log f o G Chapter 47 Theore 7 Let G be a rego C ad let f ( G) Suppose Z { a, a,, a } Let be a closed A f rectfable curve G such that ad such that Z {} The f f ' ( ; ) d a π f Corollary 73 Let G be a rego C ad let f A ( G) Suppose Z { a, a,, a } Let be a f α f ' closed rectfable curve G such that ad such that Z f α {} The d ( ; a ) π f α Proposto Let G be a rego C ad let f A ( G) Let be a closed rectfable curve G such that Let σ f Let α, β belog to the sae copoet of C \{ σ} The, Z f α ( σα ; ) ( σ; β) ( ; ( α)) ( ; ( β)) Z f β Defto A fucto f has a sple ero at a f Theore 74 Let G be a rego C, let Bar (, ) G ad let f A ( G) Let f ( a) α If f α has a ero of order at a, the there exsts ε > ad δ > such that for each ζ Ba (, δ)' the equato f ζ has exactly sple roots Baε (, ) Corollary Ope Mappg Theore Chapter 48 Cauchy-Goursat Theore Let G be a rego If f D ( G), the f A ( G)

4 Chapter 5 Defto A fucto f has a solated sgularty at a f Exaples of fuctos whch have solated sgulartes Defto Reovable Sgularty, Pole, Essetal Sgularty Exaples of fuctos whch have reovable sgulartes, poles, essetal sgulartes Probles about classfyg sgulartes as reovable sgulartes, poles, essetal sgulartes Proposto Let G be a rego C ad let f have a solated sgularty at a, a G The followg are equvalet a f has a reovable sgularty at a b f s bouded ( odulus) o a puctured eghborhood of a c l f exsts a d l ( a) f a Defto A fucto f has a pole of order at a f Proposto Let G be a rego C ad let f have a solated sgularty at a f has a pole of order at a f ad oly f there exsts a fucto g aalytc ad o-vashg o a eghborhood of a such that g f ( a) Defto The sgular part of a fucto f whch has a pole of order at a s Probles about detfyg the sgular parts of fuctos wth poles Defto A double fte seres s absolutely coverget f Defto A double fte seres u () s s uforly coverget o a rego S f Exaples of double fte seres whch are absolutely coverget / uforly coverget Defto a( arr ;, )

5 Theore (Lauret Seres Expaso) Let f A ( a( ar ; Let G The there exst, R)) C \ BaR (, ) fuctos f A ( BaR (, )) ad f A ( G) such that f f + f o a( a; R, R ) Furtherore, ad where a f a ( a) o BaR (, ) () f a ( a) o G () f( w) π ( w a) + C( a, ρ ) dw, R < ρ < R (*) ad the coeffcet a (*) s depedet of the choce of ρ The seres for f () coverges absolutely o BaR (, ) ad uforly o subsets Bar (, ) of BaR (,, The seres for () coverges ) < r < R f absolutely o G ad uforly o subsets a( ar ;, ) of G Furtherore, the seres represetato f a ( a) o a( a; R, R ) s uque Corollary Let G be a rego C ad let f have a solated sgularty at a Let f a ( a) be the Lauret Seres expaso of f o a puctured eghborhood of a The, a a s reovable sgularty of f ad oly f a for all b a s a pole of f of order f ad oly f a ad a for all c a s a essetal sgularty f ad oly f a for ftely ay egatve Probles costructg Lauret Seres expasos Probles costructg Lauret Seres expasos for ratoal fuctos Casorat-Weerstrass Theore Let G be a rego a essetal sgularty of f, the for every δ >, f( B( a, δ )') C C ad let f have a solated sgularty at a If a s