80 Transactions of the Academy of Science of St. Louis 
the whole space EF, is a constant. 
The fundamental theorem of the integral calculus. It is appar- 
ent from Cauchy’s theorem (T.40) that if f(x) is a single-valued 
regular function in a connected region R, then the integral 
iE f(x)dx 
taken along a uniform contour from Zp to a variable point z and 
lying entirely within R is independent of the path so chosen. 
The integral is therefore a function of z alone. We have the 
following theorem. 
T.42. The function g(z), where 
= J Kas 
is regular throughout R and is such that 
g'(z) = f(z). 
The principle of correspondence. Together with the power se- 
ries Yo @[-4n]x” we may associate the complex series Lo” \|@ pn Ef. 
If we denote the first by f(x) and the second by Ll (£), we write 
f(x) > |] 
at si S as i. Sle) ate te to f(x). The important 
orrespondence are 
fel sli Fl) i ‘lalg = | él. 
. The radius of uniform Son ESTE Oc p of f(x) is at least 
equal a the radius of convergence p’ of itil (£),i.e.,p 2p’. 
A power series such as f(x) is completely characterized by 
one variable and a denumerable infinity of elements. Let us 
write 
g(z 
— 
| 
f(x) = g(apr, i 5 a) 
It is then clear that the corresponding function is to be written 
ALG = g(levall, lleosull, --- 58 
where g(-- - ) is the same function in both cases, in the sense 
that addition and multiplication between vectors is carried over 
into addition and multiplication between the numbers that cor- 
respond to these vectors. 
Let us consider two power series that we express as above in 
