Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

135

i. e., becomes infinite, to the order I as r approaches s. Similarly
fl/n may be defined, and turns out to be of order 1/n, i. e., to
become infinite like (s - r)n-1l. There will be n quantities 1fn,
obtained by multiplying one of them by the nth roots of unity.
These functions, mf f/n, and also fm/n = (fl n)m, turn out to be
permutable with f itself. Since two functions permutable with a
third are permutable with each other, the functions permutable
with a given function form what may be called a group of permutable functions. Given a function of order 1 we have therefore
seen how to find a function of any assigned rational order which
belongs to its group.
Iff is a function of a certain order a, whose diagonal is positive,
then we have
fm/n = (r- s)ma/n-l (rs - ),
where the function L is the characteristic, taken with a positive
diagonal. Suppose that as m/n approaches z, L(rs m/n) approaches some function L(rs z) uniformly; then if z is irrational,
we define
fz = lim fm/In.
m/n=z
On this basis it can be shown that all the algebraic calculus of
positive exponents becomes extensible to the theory of powers of
composition, and we have
fzfzl = fZ+Z
(69)
(P21 = fZZ
Further elements f/I, f 1, fo can be defined so as to be included
in a group of permutable functions; they no longer all denote
functions in the ordinary sense, but serve as abstract elements
which may be used in calculation according to the commutative,
distributive and associative laws, and serve to lead from functions which have a real meaning in the ordinary sense to others
which have also a real meaning by means of these formal processes.