91Ó°ÊÓ

Find the families of curves on which \(\operatorname{Re} z^{2}=C_{1}\) for constant \(C_{1}\), and \(\operatorname{Im} z^{2}=C_{2}\), for constant \(C_{2}\). Show that these two families are orthogonal to each other.

Show that the "cross ratios" associated with the points \((z, 0,1,-1)\) and \((w, i, 2,4)\) are \((z+1) / 2 z\) and \((w-4)(2-i) / 2(i-w)\), respectively. Use these to find the bilinear transformation that maps \(0,1,-1\) to \(i, 2,4\), respectively.

Show that the transformation \(w_{1}=((z+2) /(z-2))^{1 / 2}\) maps the \(z\) plane with a cut \(-2 \leq \operatorname{Re} z \leq 2\) to the right half plane. Show that the latter is mapped onto the interior of the unit circle by the transformation \(w=\) \(\left(w_{1}-1\right) /\left(w_{1}+1\right)\). Thus deduce the overall transformation that maps the simply connected region containing all points of the plane (including \(\infty\) ) except the real points \(z\) in \(-2 \leq z \leq 2\) onto the interior of the unit circle.

Show that the transformation \(w_{1}=[(1+z) /(1-z)]^{2}\) maps the upper half unit circle to the upper half plane and that \(w_{2}=\left(w_{1}-i\right) /\left(w_{1}+i\right)\) maps the latter to the interior of the unit circle. Use these results to find an elementary conformal mapping that maps a semicircular disk onto a full disk.

(a) Show that transformation \(w=2 z+1 / z\) maps the exterior of the unit circle conformally onto the exterior of the ellipse: $$ \left(\frac{u}{3}\right)^{2}+v^{2}=1 $$ (b) Show that the transformation \(w=\frac{1}{2}\left(z e^{-\alpha}+e^{\alpha} / z\right)\), for real constant \(\alpha\), maps the interior of the unit circle in the \(z\) plane onto the exterior of the ellipse \((u / \cosh \alpha)^{2}+(v / \sinh \alpha)^{2}=1\) in the \(w\) plane.

Show that the mapping \(w=\int_{0}^{z}\left(d t /\left(t^{1 / 2}\left(t^{2}-1\right)^{1 / 2}\right)\right.\) ) maps the upper half plane conformally onto the interior of a square. Hint: show that the vertices of the square are \(w(0)=0, w(1)=A, w(-1)=-i A, w(\infty)=\) \(A-i A\) where \(A\) is given by a real integral.