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.

334

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

The case n =1, that already served as an example in ~ 179,
requires a particular treatment; for here round a circle with the point
zero or origin as centre we have:
2 7Z
fdz ifd.
0
Therefore independently of the radius r the integral is equal to 2 in.
Around the point infinity the integral has likewise a finite value; for,
integrating along a circle with an arbitrarily great radius II, keeping
the origin on the right, the integral is converted by the substitution
1 dz'
i=-, dlo —,*e
into: j- - to be integrated along a circle of radius -t keeping
the origin on the left; thus its value is - 2iA. The value of the
integral will accordingly depend on the path of integration; along a
finite closed curve its value will be 2i7tk or zero, according as the
closed curve goes round the origin k times or not at all. The integral
z
J z is a unique function of its upper limit a, when the paths that
lead from the lower limit z0 to S do not cross a section leading from
the origin to the point infinity; it is only in the plane perforated along
this section that the values of the integral calculated along such curves
are continuous; on opposite banks arbitrarily near the branching section
they differ by the constant quantity 2iT.
But the many-valued function which is presented by the integral
is the logarithm treated in ~ 82, 5; for, 1(z) is that function whose
derivate is -. The method there employed of rendering it a unique
function by means of arbitrarily many leaves, is valid also for the
integral. For each path not crossing the branching section we have:
Zf ( -
n a t his equation however,
and thus in particular j c = (z). In writing this equation, however,
we must observe that the quantity on the left is now determinate, while
that on the right is still many-valued. What is the value on the
right belonging to a determinate path? Before we can characterise
a definite kind of path, we must adopt a definite branching section,
ex. gr. the positive axis of ordinates. We reject the positive axis of