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.

326

The definite integral of a unique analytic function. Bk. IV. ch. 1.

This right hand integral has a determinate limiting value for
' -- 0, provided the amount of the product (c - z)f() remains finite
when z = c, for v a positive proper fraction, in whatever way the
value of S converges to c. (Cf. ~ 181, 3.)
When the path of integration proceeds in a determinate manner
to infinity, we have:
f()d = Lim f f(z) dc, for -',
^0~o:20
when the point S passes to infinity along the prescribed line.
The integral is certain to have a determinate limiting value, when
in this process the amount of zvf(z) remains finite and v is a number
greater than 1. (See ~ 181, 4.)
179. One essential difference there always is, between the integral
of a complex function and that of a real function, notwithstanding the
similarity of the Theorems of last Section with those we had before:
A complex integral can be taken along very different paths between
two determinate limits s0 and Z, while for the real integral the mere
requirement that it shall be real prescribes always one path only
between the limits. The complex integral of one and the same function
between the same limits can therefore assume various values according
to the path of integration.
When we integrate ex. gr.j dz- from the point + 1 to the point
- 1 along the upper semicircle round the origin, putting:
z = cos(t) + isin(t), dc ={-sin (t) + i cos(t)} dt,
we have:
-1
0
But along the lower semicircle its value is:
- I -7t
+ 1
The question therefore arises: Under whtat conditions is thle integral
of a complex f/tnction a unique function of its tupper limit, independent
of' the path of integration? This question, as Riema n has shown,
is answered by means of Green's Theorem and the Corollaries that
follow from it (~ 174-176).
Let the unique function f(s) -= -+ iv be defined for a given
simply connected domain. Let the functions u and v be continuous
throughout this domain. Should this not be the case in certain points,