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.

~ 178. 179. Condition of independence of the path of integration.

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let us enclose them within curves and count these on to the boundary
curves of the domain which thereby becomes multiply connected.
Integration within such a multiply connected domain will occupy
us in what follows presently. Further let us put:
efi(z) dz = F(Z) -F(s,) (U+ i 7V)-( UO+i VO) (it + iv) (dx +idy),
zo Xo, yo
where U and V similarly signify real and continuous functions of
x and y.
The function U + iV is required to have the property, that in
each point of the domain it is independent of the path of integration
and satisfies the equations:
au a v u av
v + i f'() == u + iv, u + i = ifs) =i( + iv),
ox 1 x ay Vy
so that we must have:
au _U aV av
= u, - v, v, -.
ax - a' y ax Y
We are accordingly led back to the problem previously treated, whose
solution (~ 175, p. 317) informs us:
In the simply connected domain if U is to be a continuous function,
whose partial derivates for x and y are everywhere respectively u and
- v, and if V is likewise to be a continuous function for which they
are respectively v and u, these functions u and v having determinate
partial differential coefficients with respect both to x and to y that
satisfy the relations:
au av av au
ay - ax9, (y- x;
then the functions U and V are obtained, formed by the integrals:
~,y x,,y
f (udx -- vdy) and f(vdx + audy),
Xo, Yo Xo, Yo
along any arbitrary path. But the above relations inform us that
f'() is an analytic function with the determinate derivate:
a/ ) v+-1 _ _+ _av
f'v(-/= ax ex z ^( / ^ay2
It is accordingly proved: When in a simply connected finite
domain f(s) is wzithout exception an analytic function, the integral
jf(zs) d is an analytic function of z completely independent of the
path of integration within the entire domain, and its derivate is f(z).