Colloquium publications.

90

THE BOSTON COLLOQUIUM.

It remains yet to fix the sectors within which the solutions ri
can be represented asymptotically by the normal integrals. These
sectors have been specified by Horn* in the following manner.
Let straight lines be drawn from each singular point ai to every
other point and produce each joining line to infinity in both directions. A set of lines will be thus fixed, radiating from the point oo.
Let their arguments, taken in the order of decreasing magnitude,
be denoted by
01 21... ~ r ) W C r+1 - 1... r A'. -2 r
Suppose now that the argument of the rectilinear part of the
path of integration for Pi in the plane of z lies between op-l and
o. Then rji is represented asymptotically by S. for values of the
argument of x between 7r/2 - wl and 7r/2 - w+r.t
To the general solution of (6), cl1 + c2,2 +.. + cn,, there
corresponds the divergent expansion
cSI +...+ c - iealxxP 1 Cl + + -C 2+...
(13)..+cax( + Ce + +..2
Here the real parts of two exponents, aix and a.x, are equal only
when arg (ai - aj)x is an odd multiple of 7r/2; that is, when arg x
is equal to 7r/2 - w, (i = 1, * *, 2r). Suppose then that for
7r/2 - cp-l < arg x < 7r/2 - )+r
we so assign subscripts to the ai that
R(ax) > R(a2x) >..> R(anx).
Then all the integrals for which cl 4 0 have in common the
asymptotic series c1S1, while those for which c, = c,... c;_,
*Horn, Math. Ann., vol. 50 (1898), p. 531.
t In certain cases the asymptotic representation may be valid for a greater
range of values of the argument of x, as in the case of Bessel's equation discussed
below.