Question

Let B(2) i + 22 4- 2iz (a) Find the smallest positive real value M such that for every z on the closed unit disk D, \B() < M. [6] (b) A particle on the complex plane is trapped within a wall built along the unit circle. It travels from –i to e3ri/4 and then bouncing from e3mi/4 to 1. Denote by y the curve representing the trajectory of the particle. Without evaluating the integral, show how we can obtain the following estimate. [6] |S, BI B(z)dz = V2+ V2. (c) Evaluate the integral | B(z)dz, leaving your answer in the form of of a +ib for some real numbers a and b. [10]

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Answer 1

ANSWERS Let B(Z) = {+37 4- Qiz CQ The smallest positive real value M such that every z on the closed unit disk D | Bez)]< M. | Bcz)) = 1 it az 4-diz litax tipy) 14-11x+2y) √ 4² oyna (29+49 24432 Juz+7 477444444+1 4x444²+by+16 1.1Bcz)) $ 4+4+1 4+16+16 ( b) A particle on the complex plane trapped within a wall built along unit circle. travels from -í to e3114 and then bouncing from e 31114 to 1. Let y be the cuto se represing the trajectory of the particle. We need to show , | Blzydz/ < V 2-v2 7

NOW, 311/4 e D | Blzydz! < M1 (9) B By using cauche's theorm. iA 862)dz] <DB Lite) DE and B = (1,0) DB JCH)4 (4) -V+&+1+ ša • 2012 :: 1hy Blzydz| < Vatra proved 3014 (C ) 20 10 i+२९ indot it zelo J Bleidz ie iedo 4-27 elo 4-21 010 5014 | Bcoz itzelo ido 4-dielo & (COSO tising) i-| do -2 4-21 (coso tising) 2Cosoi-(& sino41 4 +2gino-2icos do -|-2 sino + T2 COSO -12 442 sing - ligos doo atih Iwhere a: 0.04496 and b= 0.6035 -Thanking you please give like it helps to motivate me to do more wook friend