05AMidterm1solution

# 05AMidterm1solution - Math 122A Midterm I Fail 2005...

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Unformatted text preview: Math 122A Midterm I Fail 2005 50pointstota1 - Name 50 Your Scores: 1‘ 2. 3.‘ 4 5‘ Your Total 1‘ (10 pts) Lei; 21, 32 G (C, Show that 131 + Zzig + '21 "- 32§2 = 2(l3’1l2 +132?)- Explain the geometric meaning of this equation fmf: /%,+2L/t+{g,v%L/1 :(awzgm +(ﬁwﬁ-2JUV—zz} :: ~2—,§;+2ﬂ1+%/./§T;+%12 +3r§3*%/§,-«§./2, «£1,112 :3 2'2"ng : 2(’g(IL+(€LIL/ 2‘ (10 13133) Find all fourth roots of W1 + Locate these points on the complex plane. ﬂc‘ Sa9[wﬁv:h 3 4+5 L‘ :3: Z 692 3‘. (10 pts) a) Describe-the image under the mapping in m 23 of the region {}z| < 1/2, 0 < Axg(z) ‘< 7r/4}. Then sketch it on the complex plane. 1)) Sketch the following subset of the complex plane: {2 : |z — Bil a 3 a) wave”, e :fe 7/2, [a aayc.£ Merci: (‘36 4‘ (10 pts) Using Cauchy-Riemann equations, determine where the derivative of f(2) = :32 — i(y + 1)2 exists. Then compute f’(z) at those points. uzy'z’, V:__(}fljl’ axclx 3:0 l/k. :0 CWR 774m‘1v‘w3 we. caﬁfrﬂaf WZM zrﬂwléﬂj ‘ 0? yea-94”). H2) mm w/w/a x: “Cy/H), {7%) "L “x-w‘t‘vx = 2x N am We 5 (10 pts) Let f (3) be an analytic function in a connected open set A, and the axgument of f{z) is a constant. Show that f is a constant ﬂmgi rah/{ﬁe ﬁe) :: V007] [649% 1’ £5.49) cog-1c [1/ WW 6’ 1?: mhmi‘ 57:40: a? 07! 759/ 7,; czasfaj~. 5'0 (1,; sh, M3 (1/: r), 5659 (m m; 4* “4 WM JWW’J‘M‘ ‘3" “IV “#47 w (ff? 77443754; “mg 2”” H M3 2: Y7 \$7.49 V} me = « Yr me w) (n1 H’L) me 0449 =: 0 7 h n yr 5 r7 73" C7 , 0?, {3.5) 6'0 Wk“? 240W (/x 54;; TH? ‘71 WW (Iv/WM y’le [1x “Hf, r47, 3% W (Aug, we@ﬁ%};mt Bonus Problem: (5 pts) Suppose that f : (C —> C is continuous and that f (z) = f(2z) for all z E (C. Show that f is constant on C. Wmft fiat}, “"5 ﬂlz') WC ...
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## This note was uploaded on 12/27/2011 for the course MATH 5B taught by Professor Rickrugangye during the Fall '07 term at UCSB.

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05AMidterm1solution - Math 122A Midterm I Fail 2005...

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