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.

364

Expansion of ambiguous analytic functions.

Bk. IV. ch. IIlo

regular without exception the function can be expressed by Taylor's
series, and that the expansion is generalised by the method developed
in ~ 189 in case singular points occur in the domain, it still remains
to be shown that an expansion is possible also in the neighbourhood
of a branching point and how it proceeds. Then the values of the
derived functions in the branching point must be determined. Lastly
it has to be shown how the analytic expression of the function is
modified when the ramification is also an essential or non-essential
singular point.
In transferring these theorems to the ambiguous algebraic function,
the further question specially requires an answer: whether each of its
critical points is a ramification. As it will appear that this is not
always the case, but that the critical point may be merely a multiple
point without branching, the question arises: what are then the
values of the successive derivates; a question that for real values of
the function was already (~ 60) answered in the simplest cases.
194. In our investigation of an ambiguous function f(z) in the
neighbourhood of a branching point a, at present supposed not to be
also singular, in which m branches of the function are connected so
as to form a cycle, we shall set out from the consideration of a
definite integral, in accordance with the method we have always
employed hitherto. The analytical operations can be geometrically
elucidated by constructing round the point a, instead of the single
plane of z, a Riemann's winding surface of the order m- 1. This
consists of m leaves cyclically connected along an arbitrarily drawn
branching section; superincumbent points in these leaves represent
the same value of a, but to each of these points is always uniquely
coordinated only one of the m values of the function. A closed curve,
ex. gr. a circle, required to surround the point a, must be constructed
so that starting from a point S of the first leaf it is drawn round the
point a in that leaf, but then having reached the branching section
it enters into the second leaf, thence after a complete circuit round
the point a it enters into the third leaf, and so on, lastly into the mth.
From this at the branching section it returns into the first leaf and
completes its circuit there where it began.
Substituting in the function f(z) for z the new variable g connected
with z by the equation:
1
ni = z - a, and therefore, extracting the root: == (s - a)m,
let us begin with some one of the m values of this root belonging to a
determinate a, then while z describes a circle round the point a, the
value of C will change continuously with it; C however will not have
resumed its former value on z completing a circuit, but will have changed