The discussion focuses on finding fixed points for the complex mapping defined by the function \( W = \frac{z+2}{z-2} \). Participants detail the process of solving for \( z \) by rearranging the equation to \( z = \frac{2(1+w)}{w-1} \). They emphasize the importance of separating real and imaginary parts to derive equations for points in the \( w \)-plane corresponding to constant real and imaginary parts of \( z \). The method for identifying fixed points involves substituting \( w = z \) and solving the resulting quadratic equation.
PREREQUISITESMathematicians, students of complex analysis, and anyone interested in understanding complex mappings and fixed point theory.
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.ob1st said: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