MHB Fixed point for a complex mapping.

[ob1st]

W= z+2 /z-2 drawing mapping find image in w plane line Re(z)constant and im(z)=constant find fixed point from mapping

In my textbook have just W = z-1 / z+1 .

Thank a lot for your help.

 

[Opalg]

W= z+2 /z-2 drawing mapping find image in w plane line Re(z)constant and im(z)=constant find fixed point from mapping

If $w = \dfrac{z+2}{z-2}$ then $w(z-2) = z+2$. Solve that for $z$ to get $z = \dfrac{2(1+w)}{w-1}.$ Now let $w = u+iv$, and find the real and imaginary parts of $\dfrac{2(1+w)}{w-1}$ in terms of $u$ and $v$. That way, you can find equations for the point $(u,v)$ in the $w$-plane corresponding to the lines Re$(z)$ = const. and Im$(z)$ = const.

To find the fixed points of the mapping, you just need to put $w=z$ and solve a quadratic equation for $z$.