I am trying to work out what the following conformal maps do. The motivation for this is to know that what must the characteristics of lambda be so that resulting image is in the right half plane (we are investigating these maps in order to show the real part of something is positive), but I don't know where to begin!
\begin{equation} M_{\pm}^1 (\lambda) = \frac{1 \pm \lambda }{(1 - \frac{1}{2}\lambda^2)} \end{equation}
Here is an example of one of the maps. I have considered splitting it into partial fractions to get the addition of two mobius transforms. But I wouldn't know how to consider this!
Kindest regards, Catherine
Let $M(z) = (1 + z)/(1 - z^2/2)$; the other mapping is $M(-z)$. We have $$\operatorname{Re} M(e^{i t}) = \frac {2 (2 + \cos t - \cos 2 t)} {5 - 4 \cos 2 t},$$ therefore the image of the unit circle lies in the right half-plane, with the exception of $M(-1) = 0$. Next, $$\operatorname{Re} M(u + i v) = -\frac {2 (u^3 + u^2 + u (v^2 - 2) - v^2 - 2)} {u^4 + 2 u^2 (v^2 - 2) + (v^2 + 2)^2}.$$ The denominator is positive unless $(u, v) = (\pm \sqrt 2, 0)$. The numerator gives an algebraic curve which has one unbounded component with the asymptote $u = 1$ and one bounded component in the left half-plane. The sign of the numerator changes when crossing the curve.