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# 09fa-answer - Winter 2009 Math 122A Final Insmlctm‘ G Wei...

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Unformatted text preview: Winter 2009 Math 122A Final Mmch 28, 2009, Insmlctm‘: G Wei ‘ 100 points total Name Your Total 1 - n .. ‘ .. M. 1,3... 4. M 5 ‘. .6 H7. “8.. 1‘ {\$0 ptg) Evaluate the following integrals. ") Jm=1 321—; dz .v a 1' m H. r I jig“? W Aha/21% 5514! WI) {Mi/val f’ a» 1' I . {PI aha/g ’ ﬁﬁwﬁﬂf MM : I f “(‘E' 5“ 0 3 h ’i 3’ ﬂ '7’ b) me .fj'i' 43" 671w?! m (my? {fur/tréz {2" 5 ,fwvwd'ia fin; afar/{wwva } I f “Mb/(TU 4% 7' 27H g " lilgl i R 3‘30 CHM—5% 3 ‘ - ﬁJ/iz 3 “W 6 :1 ZYTL‘M) : 217 (am? away Maj/VJ ML) (1) Jéz+m=1 ark 8?: I: it M 4:46] 0’14 (Mi? ﬁt (Haj :' .. 7. . ... ‘ 3 { g 06% mm 2'2, T: [E13, glf’l IA ﬁnmj/LLZ [é W. H? Wu}? . r’e . “7!” 4‘53 A5; g :H: J9:¢[ 6 6.69:9 D 0 g) KL! COS d3, whale 7 is the line segment 301111119; 0 to w + 27', 50133— W M}: MVM’M 2mm:— w/ﬂ 724 érmyg 2, 9’, WM 5” ?_ g g % ‘W—Ht" f“). L1”: 1?: lfln‘ﬁf “:9fo” [%+L‘) o 2» (10 pts) In each parts (a) (‘13), i£ z = .7: 4» ﬁg and f(z) is as given, determine where in {he 3-piane f’(:) exists, then compute f’(z) at those points 21) M2) 5 (9:2 +231) - ihj + 3) 13) HS) "—” M2 L d1) U:X1”Xi’ L/Kzzx-{g/ MJSX Us-fo‘w) , (/x :0, 1/7:-—( 3‘ {10 pts) Let. f be an entire function If there is a. constant M > 3, such that. f(:}] 2 M, Show that f is a constant, I ﬂu [f'(%)/3M>ol 1C(%)={=O 011 (d. jpz} ; /{2j 1L1 84\$:4 fir/<16 ff?) 1?) 641/!144 {/ﬂ) 4327. f M 4.; J— / [gﬁjf :: w M (my. {,HEJ/ aM‘ m M j, W «a Md“? ff; [mm/g 3 (.24; jg} 45) R COM-ani; {:{i'} : 31%;; “"1 4/10 5‘ [Nb’b’h/‘hf. £1 (10 pts) Find the image of the set {a = a; + :1le g :1: 5 1,0 g y S 1:} under the mapping m e: ,{oﬂVWL/vlﬂ : 5.u0pw)mnfb)me“ a) Does theIe exist a. 20 so that “30) = 0? If 50, exhibit such a. :0 in the form :0 m 3:0 + iyo. If not, why not? 13) Does there exsit a :0 so that f(zG) 1-” 1? if so, exhihit such a. 30 in the £01211 :0 = :30 + 72;er Elf not, why not? + €%#O nﬁ‘ fro ‘1EKH? Ea {’0 07’? ‘ 741“ 9:0 L) a ‘5’! £0 a :1 3ft Mina/ﬁr 6 (10 pts} Let :1, :2 62 (C, Show that '31'5' 3212 + '3; “‘ 32'? = 2(131l2 “i” ‘32P) Explain the geometric meaning of this equation [y‘wojl‘r [7?! 1L YEW/11L ("gr hsz (ilﬂﬁ {71)(‘27 {5%) 7L {}I"22)(:::; ,1 F. a L r; ﬁg"; 1” 3%, ‘f’ 3L2, 97 3L 31 7L 2/}, *EP/a—rﬁfﬁ f 23%: %»'?r ;: 2({2~,(L+ {tsp/1‘) 2! .~.. “MEM‘ﬂ ﬁimw'r’wg/ﬂ r ﬁr fwd/effku I'M/w 63% {VIM—L :3/ ﬂygfqﬂ/V; h 6/”,th ﬁg: ﬂbﬂy of {find-1‘4. 7[ 545% f7 (4 ﬂ 7. (10 pts) a) Give a. precise statement of Cauchy’s inequality. 13)‘ Show that if Hz) is an entire function such that MS)! :5 Clzlw, {01' some constant C, than f(z) is identicaily zero. 09 [/3 (,R £44 75/LG 55/061 [%-h/ 2% by! /% 5 MR m 6R I fa?) fbﬁ'Lﬁi/L/A/t" z)": zusz //ﬂ LUKE CR fﬁb‘w I /%'{”’{2~o){ s: :7! LR a.) f Hz) is also analytic, prove that _f(a) must be a constant in D. b). If |/(3)l is constant, pxove that Hz} must be a constant a) { (2:) a», u + r v {M {a ) q 6/,r : a} x V , ( :5 a” 11/ '5; Wye/- " w W 7!?) (I m/Ag/q iaxw‘y “mt-tr id?“ “1’ 3"” 0', (/X Smo %/{&j ? art/Vs 3:“0 7M} 2*): a dwusgawt. L] [ﬂaw 5,0, 7%!» %‘{»2/ 3,0 rt?) 4 dad's/1’;de 5 It”) "1: “ﬁg; (MWE/FM'. (yr/50" 5; 4) 7%) 2'4 a misﬁts Bonus Question: Let f(:) be an enthe function If the real part of Hz) is positive for a1} 3, show that H3) is a constant Mail'kf ﬂwor 1: 6-4! C/ .. ﬁrm U 26’ ‘ [jg [fﬁ’th/ﬂ/LIJ/ IfIKIrq A Jagufqlzj a j i ...
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09fa-answer - Winter 2009 Math 122A Final Insmlctm‘ G Wei...

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