Question

1. if the real part of an analytic function, f(z), is given find the imaginary part, v(x, y) and f(z) as a function of x. (step by step)

2. Evaluate the following complex integral (step by step)

1. If the real part of an analytic function, f(z), is given as 2 - 12 (x2 + y2)2 find the imaginary part, v(x,y), and f(z) as a function of z. 2. Evaluate the following complex integral:

Answer 1

1) uzby Du (any. (25) – 6-) 26+7).2 (+234 au - City) (-2y) — 6²3) 2 (377) (24) . (x + y) 4- ?na x2 an za 205 405 zn 08 ou no ' 0,64,9 = (*),co shine 4,09,9)= (y) = 0 By Milnelo Thomson method, and volant f(x) = ( [0,(3,0) - 1 2(30)] d at Ci = { [-2 - 3 - 1 x 0J dz + ci = -2 SiO2 + ci t = -2 ( 222 ) + ci - Ž+ci Wagubal} .. -222 Zz+ci

f(2)=utiv f(3) = 1 + ci LEO: - Tétinja + ci ben szaitei GE) - (2ny i) (x²72)2 (2wy if x=y_rnyi (2232) 27 402 x² - y² anyi + ce (2742)2 SO' 1X,Y)= iz-c cis constant, s.

dez ; 22 let f(3) = 22 (241) 121=2 Singularation of the function is 2=0,0 = 1 2=INT #l o, Il are lier in 121=2 The function har pole at o of order 2 it abrotin Then, Res [f(z); al lim of [2². f(2)] ZOZ - z>o Iz L 27 - lim (2+1)-O- 1. (2Z) Zo Z²+1) 2 Z7L Ayesis, Reoffe ; t = ling (2+1) fee) leht 2-1) 78+) (z_) = limo z (241)

Ren [fea; - 1] = Lom (2+i) f(3) = Lim (2+) 7 (24) 21) Now (zl=2 = = = Reno [fc3); 6] + Res [te); 3] + Renife; -;] 0+2 2 - ² 0 [Answen] wen