Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

121

The functions K and K' we shall speak of as reciprocal; they will
be permutable. If now, we write
(37) K = 1 -jK(rs), K= 1-jK'(rs),
equation (36) takes on the special form
(37') KK = 1.
On account of the permutability of K and K' we have also
K K = 1.
Let us, by putting a bar over a function, denote the function
reciprocal to it. The formula for division in general may be
expressed simply, in terms of this notation. In fact, we have
_1 1 1 (1
uu + jU(rs) u 1-j - U(rs)
(38)
Let U1* Urn be continuous functions of r, s, permutable
among themselves or not, and let U1, *.., Um be the reciprocal
functions. Then the functions U and, dened by te equations
jK(rs) +- jU1)(1 jU1)K. (1 jU)( - jrs)
we get the formula
(38') = 1 (1 s) 1 - j ) ( -U
where the, q are arbitrary integers, pfunc tions o r negative, are
among themselves or not, and let U1,, Um be the reciprocal
functions. Then the functions U and V, defined by the equations
jU = 1 _ (1 _- jU1)Pl( - jul... (1 _- jUm( - jumYq
jv = l - (I - jUm)q(1 -_ - j.. (1 - jU1 —jUp,
where the pi, qi are arbitrary integers, positive or negative, are
reciprocal functions. In fact it may be at once verified that we
have
(1 - jU)(1 - jV) = (1 - jV)(1 - jU) = 1.
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