Question

if |z|=1 proove z-1/z+1 is purely imazinary

Answer 1

@myko can you do this one? I don't know how to go about it.

Answer 2

sorryy i donot know thats why i m askng

Answer 3

would making z-1/z+1 = (z+1-2)/z+1 do anything productive?

Answer 4

noo noooo we just have to prove this

Answer 5

then we could make it 1-(2/z+1)...

Answer 6

ok, lets take z= x+iy, then
\(\frac{x+iy-1}{x+iy+1}=\frac{x-1}{(x+1)+iy}+\frac{iy}{(x+1)+iy}=\frac{(x-1)[(x+1)-iy]+iy[(x+1)-iy]}{(x+1)^2+y^2}=\frac{x^2+y^2-1+i[(x-1)(x+1)]}{(x+1)^2+y^2}\)
now since |z|=1, it means \(x^2+y^2=1\) so you will be left with juast purly imaginary number

Answer 7

@Peter14

Answer 8

looks right
I trust people to multiply right, and it looks like that works.

Answer 9

just a litle typo, in the last fraction switch places (x-1) and x+1) and put - sign in between them