Question

Q5. a) Let f(z) be an analytic function on a connected open set D. If there are two constants and C, EC, not all zero, such that cf(z)+ f(2)=0 for all z € D, then show that f(z) is [4] a constant on D. b) Evaluate the contour integral f(z)dz using the parametric representations for C, where f(2)= and the curve C is the right hand half circle 1z| = 2, from z=-2 to z=2i. [4] c) Evaluate the contour integral f(x)dz using the parametric representations for C, where z-1 f (2) - and the curve C is the semicircle z 2e" Sos 2). Z

Answer 1

on analytic D. is 597 Here f(2) is also ੧) analytic. Thus is also analytic. u-iv Now f+2) = =utbir fin- candy Riemann ra it satisfies ca Since is analy tic du ou la du av e creation dy du MY § (2) is analytic a dx ㅋ independent of in defendent of =) V = constant u is constant on The food D is Constant function |Z1=2 Thus 17 J Here At each point of the rith function of this the art desierto Contour

of the frinciple branch le 1 / of beger dog? of Logz. is Z antiderivative Hence dz = Loggi -kog (-21) Z ㅈ T A 2 = i T [ N Logz log/z l tiangz not This Log se! = log 162 + 1. A Log (-21) = log2 ti 2

Z - ZP Z -1763 - Now put z zelo dz = zielo Thes 210 AP ولم 4 Te rogin it -it 2 [et*.*] 21-83 - t M V! - [ Ary)