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.

366 Expansion of ambiguous analytic functions. Bk. IV. ch. III.
1
taken along the circle with radius rm round the point zero; since
dZ = in1121-ida.
According to the fundamental Theorem proved concerning the unique
analytic function, this integral, and in general every integral taken
along a curve enclosing the point == 0 and containing no singular
point, has the value zero. It follows similarly, because (p)d:
integrated along the same path is zero, that the corresponding integral:
f (z) <!
rn r -
(z- C) "i
also vanishes. Hence for the ambiguous function the analogous
theorem is:
When the branching point of an ambiguous function, in which
m leaves are cyclically connected, is surrounded by a closed curve that
necessarily Wtinds m times round the point, if there be no non-essential
or essential singular point within this curve, both the integrals
ff(z) d a3d j'-~(S.,,:^+d )
along this closed path are zero.
The above Theorem led to the analytical expression of the unique
function in the regular domain (~ 184); its counterpart does the same
for ambiguous functions, only it must also first be generalised for a
domain with a multipartite boundary.
When there are singular points in the circle round = 0,
supposing first that none of them coincides with this point zero, let us
surround each such point - = c by an arbitrary closed curve, ex.gr. a
circle with radius Q. The curve corresponding to this curve (c) on the
winding surface by reason of the equation:
A = + t -+- oa ==a (c + 9 eifp)1- + a,
is likewise closed; it is contained altogether in one leaf, when the
corresponding curve (c) in the circle C lies altogether within one of the
sectors corresponding to each leaf; when this curve (c) enters into
different sectors, the other is also found in different leaves. But
always the theorem holds for each unique, analytic function y (C) and
q({). l,- 1 that the sum of all the integrals taken in a positive
*) In generald C -f-1- - d = 0 for k > 0.
( (z- ac) "??