C) CONFORMAL MAPPINGS (SESSION 3, 4) 1. Find the image of z = 0 under the Möbius transformation which maps i, [infinity] and 1 to 0, 1 and i, respectively. 2. Find a Möbius transformation which maps the region |z-i] <2 onto the upper half plane, the imaginary axis onto itself, and which fixes the point i. 3. A Möbius transformation T maps the upper half plane onto it self, and the circle |z −1 = 1 onto the imaginary axis such that the point 1+ i maps to i. Compute T. Is T uniquely determined by the given conditions? What is the image of the line Im= = 1? 4. Find a Möbius transformation which maps the region outside the unit circle onto the left- half plane. What are the images of circles |z| = r > 1? And the images of lines passing through the origin? 3 5. Show that there exists a Möbius transformation which maps the region given by |z-1+2i < 2√2, 2-1-2i|<2√2, |z| > 1, onto the interior of the triangle with vertices at 0, 1 and i. 6. Let I₁ = {2:12 - 2a = a}, a>0, and I₂ = {2:|2|=1}. Determine all values of a such that we can map I'₁ and 12 onto two concentric circles with a Möbius transformation. 7. Find a conform al mapping which maps the region between 2+3|< √10 and |z-2|< √5 onto the interior of the first quadrant.