Colloquium publications.

214

THE MADISON COLLOQUIUM.

(K) = (K, K), when K refers to the unit circle. Since the details
are precisely parallel to those in the treatment of the a-function,
Funktionentheorie, Chap. 10, ~ 8, they may be left to the reader.
We thus obtain a function analytic throughout this domain and
not admitting analytic continuation beyond it. This function
has the following properties.
(i) X(t, T) = X(T, t);
(ii) X(t, r) = (t - )2'(t, r),
where T(t, r) is analytic throughout (K) and
'T(r, r) = 0;
(iii) X(ta, r) = L'(t)X(t, 7), a = 1,.., p;
(iv) X(tp+a, r) = L+(t) e-4W-2aaX(t r), )a = 1,.., p;
(v) X(t, ra) = (L'(r)X(t, r);
(vi) X(t, Lp+a(r)) = La(T) ea aX(t, ).
If the transformation
ta = La(t), a =a 1,..., p,
is replaced by its inverse,
t' = La-(t) = Ma(t),
formulas (iii) and (v) become:
(vii) X(L-l(t), )= M'(t)X(t, 7),
(viii) X(t, L-1(T)) = M'(T) X(t, r),
M'(t) = 1
L'[L -(t)]
But if
tp+a = Lp+(t), a = 1, *, p,
is replaced by its inverse,
t' = L;-(t) = Mp+a(t),
formulas (iv) and (vi) are replaced by the following: