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.

~ 194. 195.

Integration round the branching point.

367

circuit along the boundary curves of a domain in which qp () has no
singular points, is zero. Accordingly the generalisation of the above
theorem is as follows:
When we form either integral:
ff(2)cd or: f r-/ d
Tn —1
(z - y.) -
in a positive circuit for all the curves constituting the multipartite
boundary of a domain, which consists of m leaves that are cyclically
connected along a branching section starting from the point = a, if
there be no singular point within such a domain, the sum of all the
integrals in each case is sero.
195. We can now proceed in analogy with the earlier development
as follows. First, let f(s) and therefore also p () have no singular
points within the winding surface round the point a bounded by the
radius r. With t any value within the circle of radius rr 1 round the
point zero let us form the function '( _ Enclosing also the point
t by an arbitrarily small circle, we have a domain with bipartite
boundary without any singular point. Then (~ 184):
1
The integration is to be along the circle with radius rrn enclosing the
point zero.
Hence, denoting tmP + a by u, so that u is a point on a determinate
leaf of the winding surface within the circle of radius r, it follows that:
T a. =/*/ 1 _________A ) d
l a. f (u) -2 * - / i- I i
Je - r)n a, -- (it a) qu a)M
Since the values of s signify the points of the bounding curve, while
u represents any point within it, we have
1 i
mod [u - aa] < mlod [ -- a].
Accordingly:
1 2 3
(z.-..)"'-(u-a)m" (z - a)"
and we have (cf. ~ 186):