Colloquium publications.

86

THE BOSTON COLLOQUIUM.

p+(h-1), p+2(h-1), p+3(h- ),....
The number h is called the rank of the singular point oo, and the
differential equation can be satisfied formally by the series of
Thomae or the so-called normal series:
(7) S=e -i 1- ** + a' hZ + + C +.. 2
(i = l, 2,..., n).
Unless certain exceptional conditions are fulfilled, there are n of
these expansions, and in general they are divergent. To simplify
the presentation let us confine ourselves to the case for which
h = 1. Then at least one of the polynomials succeeding P, will
be of the pth degree, and none of higher degree. Place
P = A AP + B p-l +..*,
Pn-= n —l+ n —l+
P.- = A~-lx + B.-lx-' + ",
o * o.....
Po = AoC + Bo0xP-l ',
and construct the equation
(8) Aanx + Aan- +...+ Ao = 0.
The n roots of this equation are the n quantities ac which appear
in the exponential components of the Si.
As a particular illustration of the class of equations under consideration, Bessel's equation (Eq. (2)) may be cited. Here the
point oo is of rank 1, the characteristic equation is
Aa2 + Aa2 + A2 =a2 + 1 = 0
with the roots
al= -, 2 = +i,
and the two Thomaeag integrals are
Yl= eiXX o + +..,
(9) ( D
Y2 = e-iXXP Do -- 'D. '