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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’iZhtnhﬁ)
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—:—zcos% 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 halfplane 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 {Irzy= 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 tnki’ Q *\ ~€ ...
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 Fall '05
 Stefan
 Multivalued function, fractional linear transformation, branch cuts, QMJ« «RR lwi

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