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# M1sol - MIDTERM 1 Math 410/610 Fall 2010 1 Find a...

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Unformatted text preview: MIDTERM 1, Math 410/610 Fall 2010 1. Find a. holomorphic branch of f2) = W ’szgﬁmmﬁ ‘f' 9; C Mm V“ng NAME L. L? M S? September 29, 2010 F—f '2 2 \3/ + , and compute f’(z). Explain all steps. 2 — 2 W ﬁrm It) {51‘ W€JA€9KLM¢QJR mg)?“ Wat/ax 'waﬁ‘ (g; gm Y: “‘“L WWQUZAL (f: 429\$ s. r ‘ ‘ at, ’2‘ H3 ; “L a "71% "i ~ 5 . "MM- fgw "H Um J.” QwQIx/x \ '1‘” ‘A— a"??? “g MW: " ’2~ “a: 3 wk» L m“ I % E %_ % Q “ 12- .wa , Mu Ci»; {Maw-w £1“ QM \ :W?\ 4“ 3 ML 3 *2: w L > a 1% '1 ‘ 2 x ‘_ mm w ~* } £2 *1? Y” 7 W; t {0 v Q» N; szw\\/‘} :3 M -- <2. ‘2: % ‘ 335+ (L ( M Wm *2 A. “E: e; e% «2 3C; » “2,3: ‘ E w ’k "2:: LE 4” A “2% k _w {~ 1%: {a (1” it w \ 71' E: ii: if}; ‘8. i, L 2. Find all complex solutions of the equation tanh z z i. %“ mt 6"“8 A: ﬁfe 3 3. Prove that 71.(:z:, = cosh :2? sin y is harmonic, and ﬁnd a harmonic conjugate 12 with 11(0, 0) 2 1. 32 WW W F} ’ ‘7 v I “k M ‘29: :1: ‘2ng v1 X‘WX 8mm (iii. mad/£1 “A 5m“ a { r2. ‘8} X“ ‘23 ﬂ» , 0 ﬂ m m x (A m—gjém K A" b\ m ‘ ‘ Cm L’ w W “U K: a, X E {a :‘(5 l <, _‘ ’3 h V , 's ‘ m 27 m5 / L buyr WM) VM Q. g} UK a 7;: C3 i W “Mk ’7 - aim \ :3 c I, > 0 \$3M ﬁévw m. £4»: 'Lx >6 m a 3/ ) we. a IX“ (13% ‘3,” +423, UK a: w: (:1qu K” “35> w” wra- Wm“ / V, cm W WEN” W W. X. CV) + A: (X 0" a: ffx) :2” C} .3 0 C265) (L ‘ ' ' ’W'CGW> X 1 :> 4123‘ M )5? (11,154) Pi“ Q. ) go a?" (K \ ff) “ ‘ “ﬁr WK ’9‘ wk ’\ Mimiﬁ’ “v” {w 33) z: g“ ‘" ‘ V 4‘, 4. Find all points z Where = 22 + zZ — £2 + 22 — E is (C—differcntiable, and compute its complex derivative at those points. Explain your computation. “Mg: Lam :1 «:5 m 2% Mi ‘5’"? a W Qﬁ alp‘ m F“? W 2 € “’"l 3‘ ‘3 {WM «9 '” Qw£wM E 3"“ l ' bi ‘ L/ w % {a Q» cllx‘laﬁacediﬁaaﬁvlp. af- 3:: a w x I) apply”) 7» (51+?) Jf l +3ng (MN (“‘4 k dry” I" . I I h A ‘ k l . {Wt MW :: 2-X+‘l “lit <3an ‘7; U «V’UWJ w‘f? (In 5. Consider a holomorphic function f : u + iv on C \ ((—00, «1} U [1, such that , 1 , ‘ r . y) : ~2—1n((m2 — yz — 1)2 + 4372;1/2). Prove that f’ z : ‘23 V and deduce that f(z\ :: log/22 —— 1‘ + a for an appmpriate branch of 109;. ("x f“. ﬁx {N3 ' f , ‘ ()1); x 0%“ c3}: &‘ dﬁﬁ W. ’* ‘ w -~ ; c: “L 35"“ '3: ” ’ w’ m M We KAI/Mt (%§ 03x. (5)}? C232,“ ’1. r27 2 w .«— l ‘ ~ b/ o a M q )1 @«ww3w«><aa)+8%a ‘7 ( .2: mm» ~+ M n w 4 .1; __________________ ““““ WW 1 4,“ >4 3 . WW,WM r4 ‘ p L L . 2, “L 3—" W W7”: “““ if”? “““““ W2: «1 2' (>4? '3 W *9 W X I} I; wwwywxw _‘~WA~,.M..meMW—w~ww~wu , .W‘Mu. 9......»me .... MW“ , , k ‘g l .' Z (x WFXYCZ a: MA, b3 +“X 3 “\“5(>> 3:: W W W _wm._,.‘..——«w—~m~ m WEWEEWW (‘73; 1321"" 351+ V >< 23 'L « "x’qui‘ér >H~wf WCMEJ>Q<“3“‘“£§2} % ‘ M} WW” ‘1 ‘W ”””””””””””””” ”‘ "” M”?TMWEWWW Q», l r WW m r 1 a W in U” M ‘LHV: %2-\ <X+W\L~\ Xmavi iihxgr Lx 3 0 J . n 3 1 k W “3 '7'“ 5 (w a M L’L 5 7637; x “@0378 +4583 _¢3 “L83:+7>:J‘ w vim—[ﬂwéﬁjf U W 5" 4W W?W€w€“ c m f w If" m C L "2— ‘3; \ J 7‘ M ‘0” , & ~A « C\((MW3M'\3UCMW)I> . 4.... (w) {w ‘26 M 33025»: m) €02: I M "-EEW“ w 1% TO 5qu 3(23:@ ‘ﬂ/xﬁJ-Ux i3 >4 alum E1W\ 6. Find a conformal map f which takes *1 into 07 0 into 7L, and carries the circle = 1 onto the circle — z 1. Find the image of the line y = 21: through f. Illustrate with a sketch. ...
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