Question

complex anaylsis

a one Consider the real valued function ulx,y). with x and y are real variables For Cach definition of ulx,y) below, find whether there Cette exists real-valued function v(x,y) such that f(2)= u(x,y) ti vex,y) is a function analytic in some DEC. If such such v(x,y) and determine the domain analyticity o for fcz). It such a not exist, prove that it does not exist. (1) ulx,y)= xy? - *?y (is ulx, y) = y3 - 3x²y (ii) u (x, y)= Arg(xriy) vix, y) exists, find ucxys does of

Only need help on (ii) and (iii), please answer both and cite theorems used

Answer 1

-Y (111 336 2 un - nity I su U (r,y) = tan li My = hty? V/v,y) = ( Un dy + 8(x) = 6 +861) = (y day to - balkaa 1 + $12) Vy (2, 4) =- 70') =0. - $.c. we take c=0 Then 66 (7, y) = - I ln (²+ y 2 / Hence, f/2) = u(3, 4) + (x,y). (do domain *#0 1 U (%, y) = ye zug $11) My. + Un = bny My = 34² 34² 0(4, y) = (undy na ply) = - 3 xy² + $(4) ly (%, y) = -3² +9 (= - 4y = 3x² 34² a om = 342 87=18 The 6(r, y) = x ² 344² Hence Hal= (y - 3x y ) + ? ( 23 324) analyhe whole domain.