An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 195. 196.

In the neighbourhood of a branching point.

369

we see that the derived functions f'(u), f"(u),.. become infinite at a in. - 1 2m - 1
the respective orders m,.., provided al is not zero.
But when some of the coefficients following a0 vanish, so that
the series presents the form:
11t+1 a+ 2
f(u) == ao + (u -- )"I al +- (u - a) m aC+l +f (,u - a) m a,,+ -.,
"_- W2
the expansion of f'(u) begins with the term - (u - a) )m at, and
for u = a this expression becomes either infinitely great of the order
--, or infinitely small of the order - according as m is
m in m
greater or less than A. For m = G the term is finite. The first derivate
of a function in a branching point can therefore also be zero or finite.
But, whether it be infinite or not, a number k can always be assigned such
that every derivate of order equal to or higher than k shall become
infinite in the branching point. For, the ]ith derived function begins
i - km
with the term (u - a) m. We have accordingly the theorem:
When the lowest power occurring in the expansion of the m-valued
function is (u - a)n:
The kcth derived function becomes infinite of the order!m - -in the
m
branching point a, when h is chosen > -;
When 7c = - is an integer, the cith derived function is finite at a;
All derived functions of order c < $, vanish in the branching
point a.
196. According to the process shown in ~ 189, the expansion
can also be generalised to a domain wherein there are essential or
non-essential singular points of the function f(z) and therefore also
of the function pp()); on the hypothesis, that none of these points
is also a ramification.
Let cl, c2,... ck be the singular points of f(z), the corresponding
singular points of q(() being respectively
1 1 1
Y1 = (CC - = % * * * yk, - (Ck M-,)m.
where in each case it needs only one of the mi possible values of
the radical to determine a point y that is an infinity point for the
function (p()); thus we have the equation:
2 J - t J C t - - t -' St |
(~) (Y.) (Y2) (Yc)
The integrals are to be taken round the point zero and round the
points y, so as to keep each of them on the left.
HARNACK, Calculus. 24