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Fixed point for a complex mapping.

ob1st

New member
Apr 15, 2013
1
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 text book have just W = z-1 / z+1 .

Thank a lot for your help.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,702
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 $z$.