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.

Fourth Book.

Integrals of complex functions. General properties of analytic
functions.
First Chapter.
The definite integral of a unique analytic function in the complex
domain.
177. A function f(s) of a complex variable z was defined in
~ 80 as a quantity which can be calculated from S by any finite
or even infinite number of arithmetical operations. When such a
function has throughout a connected domain, except in singular points,
a determinate derivate f'(z) independent of the differential d-== dx -idy,
we called it (~ 84) an analytic function in that domain. The two
constituents of an analytic function f(z) = u - iv are, as was then
deduced, continuous functions of the two variables x and y having
determinate derivates both for x and for y that satisfy the equations:
au av (au av
ax - ay' ay ax
Moreover it was shown that these equations taken along with the
continuity of the derivates are also the sufficient conditions that the
function f(x + iy) = -+ iv may have a derivate independent of
the differential, and thus be an analytic function of z in the sense
originally defined.
The following investigations, by which we are about to establish
the integral conception in the complex domain, deal first with the
function of a complex variable in general and then pass on to
analytic functions, whose general properties form our ultimate object.
In the complex plane let two given points z0 and Z be joined
together by an arbitrary curve of finite length. The equation of the
curve may be supposed given either in the form p (x, y) = 0 or by
means of a parameter x = (p(t), y = -- (t). To each value of t should
then correspond uniquely one value of x and one of y; further, to
continuously consecutive values of the variable t, which we assume
HARNACK, Calculus. 21