Vol.. 7, 1921 
MATHEMATICS: E. HILLE 
305 
point on the boundary of D(a) is counted part of the standard domain 
provided it is not a singular point of the differential equation and is differ- 
ent from a itself. In view of (6) and (7) we obtain the result: 
There is no zero of Wa(z) in the standard domain of a. 
Here is another application of the Green's transform. Suppose G(z) 
and K(z) to be analytic and, furthermore, real on an interval (a, b) of the 
real axis. Then there are solutions of (1) which are real on the same in- 
terval. Draw all standard paths of the third and the fourth kind, SK 2 
and S~K~, which emanate from the points on (a,b). The points on these 
paths, not including possible singular points and the points on (a,b), 
form a zero-free region for all solutions real on (a, b) in virtue of for- 
mula (7). For this statement we have only used the fact that for real so- 
lutions ITltl [wi(^) W2 (z)] = 0 on (a,b). The generalizations are obvious. 
The Standard Domain Is Covariant under Conformal Transformation. — ■ 
By this we understand that a change of independent variable from z to Z 
by putting Z = F (z) which preserves the form of the differential system 
(2), carries the standard domain of a point z 0 in the s-planeover into the 
standard domain of the corresponding point Z Q in the Z-plane and these 
two domains are the conformal maps of each other by the transformation 
Z = F (z) and its inverse, provided F (z) is regular in D (z 0 ) and F' (z) dp. 
0 there. This is obvious because Green's transform is an invariant under 
such a transformation and the base nets in the two planes correspond to 
each other by the conformal transformation. 
The standard domain is not covariant under a simultaneous change of 
dependent and independent variable. By a fortunate choice of such 
variables it is rften possible to determine zero-free domains of such an 
extent that the ^ is comparatively little freedom left for placing possible 
zeros; thus ont. is able to obtain a fairly good qualitative description of 
the arrangement of the zeros in the plane. The transformation 
(12) 
W = VS( Z) w, 
S(Z) = <G{z)K(z), 
often yields goo l "_e for investigation of the distribution of zeros of 
solutions in the n borhood of an irregular singular point of the differen- 
tial equation. 
1 This note is an tract of a paper offered to Trans. Amer. Math. Soc. for publica- 
tion. 
