Question

1. Let be a complex number and let 12=C 1.R>o be the complement in C to all real positive multiples of . (a) Show that the function 2 H 23 has a continuous inverse function, called 37, on N. (Hint: polar coordinates might help). Prove that there are exactly three different such continuous functions. Deduce that there is no continuous extension of 37 on all of C {0} (hint: if such a function existed, it would extend one of the three functions you defined. Now try to check continuity.) (b) Show that for any of the functions you defined on 2) is holomorphic.

Hi, I really need help on both parts of this complex analysis question. Thanks!

Answer 1

Solution :- number y ..d be a complex f2ec14R.20 By Liz-z? D 2. zz? has the continuous inverse function Ir & that equals to 2² Now, we will proove that there are exactly 3 distinct such continuous functions. For this we find the solution for 3 att ta ha bet z=reio (in poles format) a) reage ( you can the gooty ی و = ح gr y by (2273) a where & is any complex no. so there are only 3 distinct soofs for this ear) values of 23 (en & 2² -8. there are the three different (6). So, now hosomorphs so lets we have to prove une analytic (for any take ges= st. 19% that of 3 is distinct functions) fcesz na (cos et distin is in the form of f(2)= UCT,0) + v(5,0) is

Equations, So now, we know by help of Cavity-Rich are can check the analyticity of sers ! here, 462,0) = og 3. co solz), ver, o= 83 sin (0/3) :. Now . cos (9 je er 66-sią) + also a grè, sh (3) ſo that arb.(core=) from these four ears, da - šir3c0 (9) = + (.(5)) f or = {v}.sin()- -+ (frh.(sin )) . herce, &rw= 23 satisfies the Cauchy -Riemann Egre. herre 2 is hosomorpha Similarly, you can check for any other function shree. amarges