This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 4. ²ind a fractional linear transformation that maps the two circles { z :  z  = 1 } and { z :  z1  = 3 } onto some concentric circles. 1 5. a) Prove that every function of the form az + bm s n =1 c k zt k , (1) where b and t k are real, a ≥ 0 and c k > 0, maps the upper halfplane into itself and the lower halfplane into itself. b) Prove that every rational function that maps the upper halfplane into itself and the lower halfplane into itself has the form (1). 6. Let f be an analytic function in a neighborhood of 0, that satisFes the functional equation f ′ ( z ) = qf ( q 2 z ) , f (0) = 1 , with some q,  q  ≤ 1. ±ind explicitly the Taylor coe²cients of f at 0 and then determine the radius of convergence of its Taylor series. 2...
View
Full Document
 Spring '09
 Math, fractional linear transformation, closed unit disc

Click to edit the document details