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, 5B() < M. [6] (b) A particle on the complex plane is trapped within a wall built along the unit circle. It travels from –i to e3ti/4 and then bouncing from e3ti/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] | / Blends V2+v2. (c) Evaluate the integral B(z)dz, 7 leaving your answer in the form of of a +ib for some real numbers a and b. (10)

Answer 1

ANSWER

Ang we Given B(z)=1+27 4-Qiz @ By maximal modulos principle Sup. Yleczyly=sup LIB()}} 1z1 = 1 3 1B (z)) sup 41 B(2)13 ZED Iz\=1 Now for Izl=1, \1 +22) < 1+2 =3 8 14-2121714+ 12/2) = 6 So [B(z)) - | i+22 <a 50m= 1+22k3 4-217F 6 ta 4- Qiz 1/6(2) < 1 / 2 M = © з піl4 Le distance between (0,-1) & e la ta) = N2112 By symmetry, length of r=22 Z=r2tr2 = 2V2+r2 so/{ maade - fronde c #fde som 12+ G ath.

Il fearsal joininge snilstol Sretra s it a Z dz 4-2 iz r is line + it a z dz joining -i toe3ntily 4-2iZ &rz is line て大 6 9 tiit) (72 on 8,9 Z = xLt)+iy (+) 1 (t)t yCt)- it rati 늄 ytte-it-irat SO Z= m(t)tiy Lt) - ttif-i-t-irat) =t+littet dz=dtti (-dt-iva dt). dz=((2+1)-1) at - Z it az 4-9iz dz= = ja C1-9tli+(Bari)++1)dt le 4-lit-21-et fait