78 Transactions of the Academy of Science of St. Louis 
a=0, and define a;,)%", as a function on Eq) to E,), to be the 
integral 
1 ti tn-1 
nt f f tee f (¢*) "dt dt,_1--- dty. 
0 0 0 
As for @,,}, we need only note that by assigning it the norm 1, 
and observing that the norm of f* is also 1, the inequality 
lain] S [leral| -[all" = 
is satisfied. For brevity, denote a,,;%" simply by #". We can 
perform the integration indicated above, obtaining 
n'T' (an) 
e" = ——_—__—— = B(an, n). 
(an + n) 
For the infinite series Lox" = Do *"—", where x = EX, we have 
oMs 
x" = L Bian, ni”. 
0 
Here we have a one parameter family of regular functions, with 
the parameter a. The radius of convergence at a=0 is, of 
course, 1, for B(O, 7) =1. We will show that pz at ~=0, 1.e., at 
#=1, is discontinuous. 
From the definition 
Pa = lim Bian, n)—!™, 
We know 
Bian, n) pie \/ 2re!!2 log (a+1)/antna log a/(at+1)+na log 1/(at1)1 
an \12" fe 4 4 
nas aoa ey Nis 
and hence 
__ fa+1\4% 
= vii( ) (ae 1y: 
We have seen that p; 21=1, but from the last equation it is seen 
that limz.0p.=1/22. Hence p; is discontinuous at #=1. 
Additional properties of regular functions. 
T.40 (Cauchy’s theorem). Let f(x) be a single-valued regu- 
lar function in a connected region R, and let I be a closed uni- 
