Asymptotics for the zeros of the partial sums of ez. II

(1.1) D oo := {z E C : Ize1-zl 1 and Izi ::; I}. It is known that D oo is a simple closed curve in the closed unit disk ..1, and that D oo is star-shaped with respect to the origin, z O. If {Zk,n}k=l denotes the zeros of sn(nz) (for n 1,2," '), then it was shown by Szego [7] in 1924 that each accumulation point of all these zeros, {Zk,n} must lie on D oo , and, conversely, that each point of D oo is an accumulation point of the zeros {Zk,n} Subsequently, it was shown by Buckholtz [2] that the zeros {Zk,n} all lie outside the simple closed curve DOC' As for a measure of the rate at which the zeros, {Zk,n}k=l' tend to D oo ' we use the quantity