ABSTRACT
Although the choice of the negative real axis as boundary for A simplified the narrative in this example, it restricted the regions K,(a,) (and their z maps) unnecessarily. When a , = - co +is, the region K,(a,) can be extended by rotating the cut in the positive sense until it coincides with the positive imaginary axis. The total region of validity is then -fn < ph < < jn. Further extension is precluded by the monotonicity condition. Similarly when a , = -co-is the maximal K,(a,) is - j n < p h < < f n . Ex. 11.1 Let € = + 1 be the only finite singularities of $(a, and Y ( F ) converge at infinity. Using all necessary Riemann sheets sketch the maximal region KI(-a) . Ex. 11.2 Let aJ be at infinity and Conditions (i) and (ii) of $1 1.3 be replaced by the stronger conditions: (i) the € map of gJ is a polygonal arc; (ii) as t passes along gJ from a) to I, Re(€(t)) is strictly increasing if j = 1 or strictly decreasing if j = 2. Show that HJ(aJ) is a domain.
