Question

Let i + 2z B(2) = 4- 2iz (a) Find the smallest positive real value M such that for every z on the closed unit disk D, B(2) < 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 the curve representing the trajectory of the particle. Without evaluating the integral, show how we can obtain the following estimate. [6] || Beydals 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

the Sol:- from give data we haver i +22 let B(2) - 4-212 a we have to find the smallest positive real value M such that for every z the closed unit disk D, 18(2)| EM on 16123) i+gz 4- 2i2 2=2 tiy i+2(xriy) 4-2; (x+y) i+qatily - 4-2istry 4x + (29 +1) ✓ (2y+4)² + ux 4x²+4y²+nyt! ny²+16+ 16y +4x² Maxima of 1B1z) occmons on the boundary IB (2)| < 1+ut) 9 I 2 9+16+16 36 ..M= 137 t។ B А

Now 16 662)d 21 <ML(8) (By using caruchy's theorem) د < DB | foods (a) : 6 =(10) (1+ha) ** (a) DB = vo 1++++ 2+ ald DB = 2+52 s l2+52 is (Hence Blz)02] proved) C Now we are evaluating the integral i ſ B(2) d2 answer in leaving your the form of a rib for some real numbers a and b 8: 2 = e =eio 3tly O itseio ه io it selo io ie do ie of the do t 4-2ie a> | 8(2) dz = 4- 2ilio -T/2 3tly

f 8(2)dz - itacio 4- Zielo i do -T/2 o do - 1 2. (Coso teseno) ;-) 4- 2* (coso + iseno) -T/2 o 200501 - (25n0 +1) do - y+asino - 2icoso -T/2 -1-asino +2 i coso do a 4+2 sino - 2pcoso - 1/2 = ati6 we By solving get a= -0,04496 b =0.6035 2. If you with Answer. Impress please give me like.