Question

. :) 5 Find the maximum of lezel on the unit disk ii) Find the maximum of u a Rez² on the unit square R [0, 1] x [0 iii) Find the harmonie conjugate of U(x,y)= et cosy 5 50 iv) Find the Isinal maximum of [0,25] x [0,25] on

Answer 1

(6) Co For a soln; let 412) = 872, ZEE 2 1 1 2 1 =1} since t is analylic on & IZJ12141, by maximum modulus, theorem, for ZEIZI IZL412 f attains its maximum value on 121=1. Now polar form of z (1210 =1) is = e io, ocoLLIT Then for z =010, locoL2T), lezy=le ezig le cosso tisinco - le cosol - lezy - e cosso Therefore maximum value of lezy on the unit disc is = e cosos (coso is maximum for 0-0 in the settelo 506211) =e (ii) ( U =ReZF = Re (atig 3 = Re (23tzning tuningriya) → u = 23 - Bayt (24) E [0, 1]X[0, 1) U2 = 0 and wy=o gives 34²342=0, 62y =o. n = - = my=0 yno ory=0 Therefore U = 0, y = 0 => n=y=0 . U60,0)=0 Lul,0)=1 This shows that u cannot attain its maximum value at any interior point of toux [017 For OCaLL, y =o the maximum value of uis = ULO) For osnul, y=1 U2 = 23-32, ota 5) and maximum value = {0, 1) = 0

For y=0,022 =27 Hizl= fles_eth) -Ź Zisine) = 1 sinal which has maximum, value. I sin 71 For y = 211, 22LRT 142) = elati21-lati26| = 21 e 2t ein- & 2T etaj = { 1 eur (cosa Hisina) - e 21 (corn-isina) -21T - 2TT) cosa tice Ersin te 2T) sinal = // J ce ate 2 cash the 24 e 27 j2 sina Ź Cent72032 [ 2T +ē 2012 le 21 - 8 217) u sinh = / 2 v celta e 214 4 sinh which has maximum value - - v ce ate 20th sin 2 = 1 - v Ceutte 2772 = uttē uu) Hence the maximum value of lsing over the set T211) [211] is Hearte)