Review Exam II Complex Analysis

Transcription

1 Revew Exam II Complex Aalyss Uderled Propostos or Theorems: Proofs May Be Asked for o Exam Chapter 3. Ifte Seres Defto: Covergece Defto: Absolute Covergece Proposto. Absolute Covergece mples Covergece Examples of seres whch coverge, coverge absolutely, coverge but do ot coverge absolutely Problems about seres whch coverge, coverge absolutely, coverge but do ot coverge absolutely Lmt Iferor & Lmt Superor Defto Examples of lm f, lm sup for whch lmf a, lmsup a Problems about lm f, lm sup for whch lmf a, lmsup a Power Seres Geometrc Seres Theorem. For a let. The, propertes (a), (b) & (c). lmsup a R = = a = Proposto.4 Rato test: For, R= lm a / a +, f the lmt exsts. Examples of power seres wth radus of covergece, R, R Problems about determg for power seres the radus of covergece, R Propostos o Absolute Covergece A) Sum & Cauchy Product of Absolutely Coverget Seres s Aga Absolutely Coverget B) Radus of Covergece for the Sum/Cauchy Product of Power Seres s at Least the mmum of the Rad of Covergece of the Summads/Multplcads C) Absolutely Coverget Seres ca be Rearraged Assume the followg coveto throughout the remader of these otes, uless otherwse specfed: G = rego (ope, coected subset of C ) ; fgg C, :

2 Chapter 3.2 Dfferetablty ad Aalytcty Defto of dfferetablty of f at a pot G Defto of dfferetablty of f o G Defto of cotuous dfferetablty of f o G Defto of aalytcty of f at a pot G Defto of aalytcty of f o G Coway: aalytcty of f o G Defto: A(G) Defto: set of etre fuctos = A( ) C cotuous dfferetablty of f o G Proposto If f s dfferetable at a G, the f s cotuous at a Proposto Let f, g A(G) ad let α C, the α f, f + g, f g, fg A(G) ad A( G Z ) f / g \ g α C f( ) α, g( ) f, g Proposto Let ad let. The, A( C ). p g = f C q Corollary Let f be a polyomal ad let be a ratoal fucto. The, A( ) ad A( C Z ). g \ q f ( ) = a ( ) Proposto 2.5 Let have radus of covergece R = R f > = g f = a) The, has radus of covergece R R (Actually equal) g( ) = a ( ) b) f A(B(,R)) ad f ' = g c) f s ftely dfferetable o B(,R) ad! ( ) ( ) ( k) f = a = k( k)! k d)! ( ) a = f ( ) Proposto 2. Let f A(G). If f ' o G, the f s costat o G.

3 Complex Expoetal Fucto e Defto Dfferetablty ( e e ) ' = e = ee e + w w Re t Propertes,, e = e, e = cs t (for t real), Perodcty, Complex Trg Fuctos x+ y x y x e = e = ee e = e,arg e = y, Euler s Equato, e = ff = 2 k for some teger k Defto cos(), s() cos ' = s, s ' = cos Dfferetablty ( ) ( ) Propertes Pythagorea Theorem, Addto Formulas, Represetato Formulas betwee Complex Expoetal ad Complex Trg, Perodcty, Zeros Other Trg Fuctos ta, sec, cot, csc Complex Hyperbolc Trg Fuctos Defto cosh(), sh() cosh ' = sh, sh ' = cosh Dffertablty ( ) ( ) Propertes Complex Logarthm Pythagorea Theorem, Addto Formulas, Represetato Formulas betwee Complex Expoetal ad Complex Hyperbolc Trg, Represetato Formulas betwee Complex Trg ad Complex Hyperbolc Trg Defto of a brach of log o G Implcato that G Proposto 2.9 Totalty of braches of log o a rego G Defto of prcpal brach of log o C \(,] log = log + Arg where < Arg < Proposto 2.2 Dfferetablty of a cotuous verse Dffetablty ( log ) ' = (for ay brach of log ) Complex Powers Defto b = exp( blog ) va prcpal brach of log ' = b b b Dfferetablty ( )

4 Cauchy-Rema Equatos Let f = u + v. The, f A(G) ff the partals u, u, v, v exst ad are cotuous o G ad ux = vy vx = u y o G. x y x y Proposto Let f = u + v. Let f A(G). The, u ad v are harmoc o G. Defto: Harmoc Cojugate + C \{} C \{} 2 2 Proposto log x y s harmoc o but does ot have a harmoc cojugate o Proposto 2.3 Let G be a dsk or the etre complex plae. If u s harmoc o G, the u has a harmoc cojugate o G Mappg Propertes of Stadard Fuctos e : G C where G s a rectagle of the form G = { = x + y: a < x < bc, < y < d} G = { : Im < } G = { :< Im < } G = { : Im < 2} G = { :< Im < 2} Ex : Ex 3: Ex 2: Ex 4: : G C where G s a sector of the form G = { = re θ : a < r < b, c < θ < d} Ex : = 2, 3, 4,... ad G s the frst quadrat Ex 2: = 2, 3, 4,... ad G s the upper half-plae Ex 3: = 2, 3, 4,... ad G s the rght half-plae Ex 4: = 2, 3, 4,... ad G s B(,R) frst quadrat G = { : Re < ad Im > } s : G C where G s the base half-strp for s,.e., 2 log : G C where G s a subset of C \(,], specfcally where G s a sector of the form G = { = re θ : a < r < b, c< θ < d} Ex : G s the frst quadrat Ex 4: G s B(,R) frst quadrat Ex 2: G s the upper half-plae Ex 5: G s C \(,] Ex 3: G s the rght half-plae : G C where G s a sector of the form G = { = re θ : a < r < b, c < θ < d} Ex : = 2, 3, 4,... ad G s the frst quadrat Ex 2: = 2, 3, 4,... ad G s the upper half-plae Ex 3: = 2, 3, 4,... ad G s the rght half-plae Ex 4: = 2, 3, 4,... ad G s B(,R) frst quadrat Ex 5: G s C \(,]

5 Coformalty Chapter 3.3 Defto: path, smooth path, pece-wse smooth path Defto: taget drecto to smooth path at pot where γ = γ () t γ '() t Defto: agle betwee two smooth paths at a pot = γ ( t ) = γ ( t ) where γ '( t ), γ '( t ) f γ, γ2 G = γ( t) = γ2( t2) γ'( t), γ2'( t2) Theorem 3.4 Let A(G) ad let G. If f '( ), the f preserves agles for ay smooth Defto: Coformalty paths such that ad B-Lear, Lear Fractoal, Moebus Trasformatos Defto ( ) a + b S =, ad bc c+ d ad bc Dfferetablty S'( ) = 2 ( c+ d) Propertes : C C, S s -, S s oto, pole of S s -d/c, ero of S s -b/a S d+ b d b T( ) = S ( ) = =, S coformal o C \{ d / c}, Fxed Pots, c a ca+ a Uqueess Specal Cases: Traslatos, Dlatos, Rotatos, Iverso Every B-Lear Trasformato ca be wrtte as a composto of Traslatos, Dlatos, Rotatos, ad the Iverso Mappg Propertes of Specal Cases: Crcles to Crcles Geometry of Images