HI

i want the answer of the highlighted questions it's from complex analysis subject

id Perm Frat Matherubio Dejurthe First Mid Term Exam 487 Math Arqwer all the following questions Question1 Brown or dinprove the following statements - Argla + 20when Bar =0 b-f[z) = Arr is continous at : = 0 CThe wequeree [Argt-1 + 21] is servergent to0 Question? i-thow that if two analytic functions luive the sure derivative throughout a domain D, then they difler only by an addition constant. - Find a positive integer n for which the equity holds c-Find the domain of analyticey of : (() -2{1 -y)+ix"-y + 2y) d-Evaluate ( Question 3 a Show that if fau+is is differentiable at za, then it satisfies the C-R equations there. b-Let (k)- Show that f satisfies C-R equations but it is not differentiable at J-O.7 91 * P W1 :TO STC IIn P 1:IA : Y - 19/T/1) King Saud University College of Science [Girls section] nahan First Mid Term Exam 385 Questioni Proareor disprove each of the following statements ajlf f is continuous at Ep then Imf is also continuous at zp- b) Im() is harmonic in any domain not containing the point z=0. c) Arg = Argz - Argraz # 0.21 0. Questionz a]Show that (itsimsstoney" cozn (# - 6) + isin = ( - 9) b)Evaluate[ 342 c)Find the domain of analyticity of / (s) = Is| + z. Question 3 a]Show that if /(z) = u + iv is differentiable at zo then it satisfies The C-R equations there. b] Let / (z) = 0, z =0 Show that f satisfies C-R equations but it is not differentiable atSTC IIn. P 1 1:14 . Y . 19/7/1) King Saud University First Semester College of Science 1437/1438 Mathematics Department [Girls section] First Mid Term Exam 487Math Question1 Prove or disprove each of the following statements ha)Re(z, 32) =|z llz;] if and only if 6, - 03 - 2nn. n d N. b]There exist an analytic function frutiv such that u = x' + yl , T de)Airg(zz) = Arga-+Arg(za).2 0andz; .:0 F 1-020 d)la +22l' + 12 - 2;|2 = 2(]z,|2 + 1zal?)v2 12geC-7 Question2 wowrajFind a positive integer n for which the equality holds: T.-& T 3 (-1410"=1. C hiSolve the equation: (z + 1)5 = 25 If feutiv is analytic in a domain D show that: Question 3 lajShow that if f(z) = u + iv is differentiable at zo then it satisfies the C-R equations there. b] Let f (z) = rlyJeuxlyi 2+0 Show that f satisfies C-R equations at 2 0 but it is not differentiable at 2 0