Zeros of functions on the complex plane

[zetafunction]

what is the relationship (if any) of the following statement

- A function has ALL the zeros on the line (complex plane) $$Re (z) = A$$ for some Real A

- A function has ALL the zeros on the unit circle defined by $$|z| \le 1$$

i think there is a transformation of coordinates so the line Re (z) = A would become the unit circle but i do not know what is.

 

[skook]

Think it's f(z)= (z-A+r)/(z-A-r) where both A and r are real numbers. The line has inverse points lying either side of it and mirrored by it. The unit circle has its centre and inf as inverse points. So you can map the A-r to the centre of the circle and A+r to infinity, then scale the circle to the right size by mapping A to 1. It works because the line is a generalized circle.
:-)