MATH 227 Exam 2, Spring 2018

MATH 227: Introduction to Complex Analysis Spring 2017-2018, Midterm 2, Duration: 60 min. + SO un - Name: AUB ID: Exercise | Points | Scores 1 15 2 25 3 15 4 20 5 25 Total 100 INSTRUCTIONS: (a) Explain your answers precisely and clearly to ensure full credit. (b) No book. No notes. No calculators. Exercise 1. Compute $ f(z)dz where f(z) and ¥ are given below: " e=3 (a) (5 points) f(z) = ———————- and: |z| = 1 =fap; ) points) (2) = Fy dy Per ' ' {@) las feo simfle ples, 2-2 ad 2a “ > bu onl Fes wud ged wy 2-5 S-g he — dz = 27 € : (2-%(2-}) 2-2 Ie 4k “ha <1 2 = AR L-@ 2 -_ iw 3 (b) (6 points) f(z) = MGB) andy: |2 i = 3 iz(z —i) Kole Kal ats) “* 2) halamenfice ingide Y and BM Xt (2°) secon} ay Biuple ple Zat ( Ge de = Que ame fe-de 1 12 2 2 ai kk (Leg i ) = RH niCE) akTi ei *(F) = QT (c) (5 points) f(z) = alia ed == andy: |2|=4 23(z — 7) as Fe f@ lat & Bacple re at Z-T a exovable singalaces at z=eo dinle , foe ‘ghese | open g, , £Os2 - aT 2- Gr2 2> ‘al =4 227 ~8ro TOME as = t & T ad (s Bila lz a 3! S! z = Lu «= £2 #4 3! st s i ee ne of are Exercise 2. 1 (a) (5 points) Compute the Laurent series expansion of f(z) = z cos— for z € C*. Zz Zl L = Ss en Z r 2 X20 @u) . =" Gr)" yee IE: —_ ' "* @)! i aig Tae (b) (10 points) Let a,b € C. Show that f(z) = ———————~ admits the Laurent (z—- ae — b) series expansion a 1 b Gfz-a\" f(z) a—bz—a (a wo ( 2) , for0< |z—a| < |b-al. LQ) - oo” |@-) +t | SE ~ (1+ f2)-G8) & a foc era] <\b-s| - = (22) Ct _- ee (e-a)" n=o Cb- = “b 2-0 et Goa)" "nS Gay! >, 1} a SF &)" © a a" - ey ee - AS Jj 2&9 Rot (c) (10 points) Show that 1 ~ 1 cosh (- + *) =dgt+ > Gn G + =) ; for all z € C*, z 1 Ze where 1 2a An cos(n@)cosh(2cos0)dé . on Ih Wo age Ke defi Men of Lause “ys ey, eosh(2 + Ly); 2 a, 2" where Z, = 4 cosh(ia 4 ') deo we “epee | wee ‘ to . m (ay- | co =S | wali ws €& ) Wes = Qe e€ (ot) dea =idée ex we oy = woth (Veor6)e do oT “e n=—-@ uz 00 e “ 2 a Ce Z Qa 42 = y a, = h=-co Mz oe : oa - Le [.. - al 21s (49) ah (2eu06) de (aot i guste) Le) = LE 2) => yyy, = 0 WweZ 6 Exercise 3. (15 points) Let f(z) be a holomorphic function with k distinct zeros 2;,1 = 1,...k, of order m; respectively. Show that f(z) = (z — 2)™(z— 2)" ...(z — zx)’ g(z) for some holomorphic function g such that g(z;) 4 0 for alli =1,...,k. Gc Zz, & a 2e0 of ocdes my Here ecists of helewartlie eolhe 3a) 40 - te if! 4@)= @-2)"3.@) Noo? ay * iS A 2eLo of order my for 41 (2) Quee. Fes)- oe leo= 4 / m,~i aud f(z)- m, (e-2, 1) + (2- 2) x “@) (me) . > ; : { @zre > 51 G) fo ty | Kae ewe tase wmode use ef Ve fact ef eeu Vir' Howe Ce 2es0 of atlas my fer g (e) 1 Tees Pee wats $e Wemaglie Pe 3: @) 4 shel Veet Gr) = @-%) e 422) Re perhede- ve. le) 2 (2-2) (e-2* eG) flap en bee weet: how Krak ge @) has keo rems BH dyke of onder wey volt @)= @- a bo Hewee, $0 = (2-2 )"@-%)* we) fos bom § bol. aud sore Verse lee Exercise 4. (a) (6 points) State the maximum modulus principle and the minimum modulus principle. Boclusode (b) (4 points) Let h be a holomorphic function. By using the Cauchy-Riemann equations, show that if || is constant, then so is h. Ad hauwrtiv , 4, rea velead RSs « ER oe a | ) hy = —Vee Koo if \W =o, Re hee cogtarct /u=0, Va. bur pose \ul #2 @o Wed} wrsvizeo - Tee Jk|> = ut av™ = CC eonthef Difeeskasiog ides bk, tVV, =O aud Way + VW = © Kilts, by ~ ex wees FUVV, = Oo and bv uve, + VV, 4 1 : 7 oa 4 a al ine wo, ae wv iy =a le vale Ay + V Wy by ce. addict => (rey \u, =O = 4&e HEO, Simla, Uy = Vee =O, Youse hy et (c) (10 points) Let f and g be two non-vanishing continuous functions on D,(0) that are holomorphic on D,(0). Suppose |f(z)| = |g(z)| for all |z] = r. Using (a) and (b) above, show that there exists c € C with |c| = 1 such that f(z) = cg(z) for all z € D,(0). Defc h(e)- we Ta fhce)Jet Gr al fen —@) xe Since ke helemor glee on D, (c) + west aad be nmeaxceuee wodedad aud Bom Condon, Qind yivruniunr veoduhes om feh=al. Ry (e\ ar wc) be ewal , wd pe dene [h@[=1 om Dey ve R@|=eh@l + Be) By (b) «ce sles lenoud Yaad hie) ec cold oo D, () | ais gree [h(edl-0 Jef et Nowee 0 (2) eS € s@ V z é0,£) abaare (ol = ( Exercise 5. (a) (5 points) State Liouville’s theorem. Boole © ase (b) (15 points) Prove Liouville’s theorem. Bodesorte (c) (5 points) Let f be an entire function. Suppose there exists a positive real number M such that Rf(z) < M for all z € C. By considering g(z) = ef, prove that f is constant. woe baw [9(2)| 12 [ Appl 4 Licerle go Reb Since ¢ &) ss etoe basudel Rurdivor must be coughed | Kaw 4, ‘(2) = =f @Be f@) £@) 0) ve. a tone ' 2 ois® < Pi foc all 2 € ¢ 5 ~~