final sol'n - 1(20 poinfis Consider the multivalued...

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Unformatted text preview: 1. (20 poinfis) Consider the multivalued function 'u) = fizz — Deter— mine its branch points with their phase fantors and draw branch cuts so that branches of this function can be defined continuously off the branch cuts. Describe the Riemann surface of the function. “3.35%”? : h—Dkz’iZ-htnhfi) i zfi- | Tc Wmléicimz (it :\,t\,‘ fie“ :e =4 1—5pm“ ~26 a: O I Q L :‘e, 5‘ * m‘ — mun—Li . it °° ‘ 0ft " M N5 -. mfg—o cwnumuim W W WW UM)QMJ« «RR lwi ed} 2. (20 points) Consider the fractional linear transformation that maps ~1 to 2i, 0 to 41 +1', and 00 to 1 + 1" Determine the image of the imaginary axis under this transformation. " . ' ‘ t \ — — ‘ Q 3&3 336w => ’53 {my} 4% 4a H v73: QM}??? 93 (934“ W) = Max = («5135,15 4mg: 37% \ ‘ t/\ \—'“( (L ‘ “KY\:L\*L3 :L\Z\(L "-1-"??? WW cs ih QM W «amt, Me‘s; wifiicswmm :. 3. (20 points) Is there a domain D C C such that the function f(z) =z+§+isinz—:—z-cos% is analytic on D? Justify your answer. % :X+C‘3 ) )kaéIR ¥Kfls ’Lx kéwx m3 :u-‘(txr M [1m ) 09' {5 Mi We 4. (20 points) Suppose that f (z) is an entire function such that f (z) /z is bounded for Izl 2 1. Show that f (z) is a polynomial of degree at; most 1. fimfifl. égfl; icing? V¥e¢ Qn- L g )(“120 .’ 21cc, gm M41 & 4 L‘ ,. 3 ‘ KM “LN—T S 33% (be 17 fl MFR I“ Mum 5. (20 points) Find the number of zeros of the polynomial p(z) = z7+22+2 in the right half-plane Rez > 0. ‘6 I 4Q ‘ W MMAWM 32m 3’ {a SM‘Mmfl a '%“}'(-g +E‘ : 4I-»g| V Y PA 95 M a V? Rm 952%! %=~Cux 3 439% R “A: Wm 44W+Ha§+z= {lg—{+1 = 4321+”? _ Pk't‘éA W Ter 1 W3“ 1‘4ka “Rg%4;fi )qutfiqu 3r 4Y7”ka 3 W: W 1Mc¥ou§iqfi jam 61mm fl L > V y -m * JR {1 “6 S R - HM “’ m > jolfiwnab 1;: - 0% 934(MP(*3}=T+€L CR [R M 275(0)“) “‘F‘K/‘V é‘+—W'\’ iL: WHO + zl‘fEL: No ’ tlJ‘Q’L \ihfi Zr ————/ L l " \idflf Q wwwagm M N ’LT '7 4m; in La ms W wwww gimp) m ma“ . Ham=\2"+2\>\2’\:\%>W%ex Ha Rm LEW mm = MM: WH 7 ‘31 sweep M e R H“ > “5' A”? W m M vim 2H >\ 23" # W) WW1 WM 7W2“ “V3 63D “’4 w W,” W ) flazgflfla Muflwm thfiufiiws “Lt. 6. Consider the function f(z) = 3% in the annulus {4 < |z| < 6}. Let f (z) = fo(z) + f1(z) be the Laurent decomposition of f (2) so that f0(z) is analytic for |z| < 6 and f1(z) is analytic for |z| > 4 and vanishes at 00. (a) (10 points) Obtain an explicit expression for f0(z). (b) (10 points) Compute f,7 f(z)dz where 7 is the circle {Irz|y= 5} trap versed once counterclockwise. {A $4 :0 e522: :1; t e7 iatkif) keZ in mm at war a: Lkrrr \ v x \ :2.» v c LL bwkfl n = ‘ fi/ ,7??? Q 'l t-nki’ Q *\ ~€ ...
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