Elementary Transformations from z to w plane

• March 24th 2008, 01:57 PM
free_to_fly
Elementary Transformations from z to w plane
I'm stuck on this question:
For the transformation:
w=(z+i)/(z-i)
show that as z moves along the real axis, w moves along a circle centre O and radius 1.

I know that since z only moves on the real axis, it has only an x component, so I can replace z with x, but I'm stuck after that. Do I rewrite w as u+iv? If so I get a very complicated equation which doesn't look like the equation of a circle. I'd be grateful of any help.
• March 24th 2008, 03:34 PM
math_science_dude
free_to_fly,

Plug in x for z and multiply (numerator and denominator) by the conjugate of the denominator, which is x + i). Then separate into real and imaginary parts, and take the modulus of the expression. It turns out the modulus of w is 1, (i.e. ||w|| = 1). Thus the image (in the w plane) of the real axis under the mapping defined by w = (z+i)/(z-i) is the unit circle.

Best,

m_s_d
• March 24th 2008, 06:03 PM
ThePerfectHacker
Quote:

Originally Posted by free_to_fly
I'm stuck on this question:
For the transformation:
w=(z+i)/(z-i)
show that as z moves along the real axis, w moves along a circle centre O and radius 1.

Let $z=t$ then $w=(t+i)/(t-i) = (t^2-1)/(t^2+1)+i(2t)/(t^2+1)$ this is a parametric equation for almost a circle. Because we never reach $1+0i$.