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.

~ 180. 181. Integration within a multiply connected domain.

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When within the domain a closed curve is drawn, (a, fig. 17), which
by itself encloses a simply connected domain, the integral along
this curve vanishes; but when a curve is
drawn which together with a boundary
curve encloses a domain, ex. gr. Af, the
integral along this curve is equal to the
~I \ ~ J \ ~value that it assumes for the associated
~/ /,_^ / ~ \boundary curve.
'\ a From this we form the rule:
3. When f(s), the function that is
to be integrated, loses the property of an
analytic function in a discrete set of
\.. ':.....points of a simply connected finite domain,
Fig. 17. by either ceasing to be continuous, or
to be finite, or to have a derivate, let
us enclose each such isolated point within a curve arbitrarily near it,
and, counting all these among the boundary curves of the domain, let us
carry on the investigation in the multiply connected domain. When
the integral for the boundary curve round any such point vanishes, the
boundary of this point can be dispensed with, although it is always
possible that the integral function up to such a point may likewise
lose the property of an analytic function. For, it was only proved
for points at which f(z) remains continuous, that the integral function
also is analytic.
There are therefore two things to be examined: first, whether the
integral round a singular point vanishes; second, whether the integral
up to the singular point continues finite or analytic in general.
We can at once see that: if the integral round the singular point
do not vanish, the integral function is an ambiguous function, and the
singular point is one of its branching points; along a branching section
starting from this point the values of the integral differ by a constant
quantity.
For if the value of the integral round one boundary curve of the
point a be A, it has the same value for every boundary curve round
the point a, because two boundary curves determine a ring surface in
which the function is analytic without exception. At the two sides
of a curve starting from the singular point the values of the integral
differ by A; the integral is therefore ambiguous.
A determinate value of the integral up to the singular point is
in this case quite out of the question; the value of it then depends on
the path by which the singular point is reached; how often ex. gr. the
branching section is crossed on the way.
But next: if the integral round the boundary of a point c vanish,