Problem,4 Verify that w = f(z) = (z? 1)1/2 maps the upper half-plane Inn(z) > 0 onto the upper ha...
1. Show that w = 2 + the w-plane. NI maps the half circle | 2 = 1,0 <<n in the z-plane to the line segment -2<u<2 in
2. Find all one-to-one analytic functions that map the upper half-plane U onto itself. (Hint: φ(z-i(1 + z)/(1-2) maps the unit disc onto U and φ is one-to- one.) 2. Find all one-to-one analytic functions that map the upper half-plane U onto itself. (Hint: φ(z-i(1 + z)/(1-2) maps the unit disc onto U and φ is one-to- one.)
Question 5. Let f(2) = for z e H4 = {z : Im z > 0}, the open upper half-plane of C. 2+i [2]a) Show that f maps H4 into the open unit disc |2| < 1. Hint: compute |f(2)|² for z e H4. [3]b) Show that ƒ maps the boundary of H onto the boundary of the disc |2| <1 minus one point. What point is missed?
(Complex Analysis) The linear mapping wFUz+p, where α, β e C maps the point ZFI+1 to the point wi-i, and the poin to the point w2-1i a) Determine α and β. b) Find the region in the w-plane corresponding to the upper half-plane Im(z) 20 in 9. the z-plane. Sketch the region in the w-plane. c) Find the region in the w-plane corresponding to the disk Iz 2 in the z-plane d) Find the fixed points of the mapping The...