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.

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

d x; consequently the equations
-au av av )u6
ay x; Cy- x a
must also hold for the points of the curve, that is to say, the original
assumption is impossible.
d) When the function f(S) is discontinuous- at the two sides of a
curve lying within the domain, the paths of integration at the two
sides of such a curve also lead to different values; i.e. the integral
also is no longer an analytic function in the domain.
4. When the domain of the integral contains the point infinity,
we convert it into a finite domain by the substitution =_ - To the
integral
-f ( '
taken arbitrarily closely round the origin in a positive circuit will
correspond an integral in S along a curve likewise' surrounding the
origin but arbitrarily remote and in a circuit keeping the infinite on
the left: such a curve is said to surround the point infinity. The value
of this integral is certainly zero when
Lim z' f(1 )- Lim {f(z). =- 0.
The integral function will remain finite even for the point infinity,
according to the theorem just proved, when we have:
Lira {mz'f() 4 -- Li1m If('). '2'5 L= m ]2-m G
'0 1' =!2-! - - G
some finite quantity, where the exponent 2 - v >1 or v <..
182. Of unique analytic functions presented in an explicit form
we have only become acquainted with rational algebraic functions and
with the infinite series of ascending positive integer powers within its
circle of convergence. These we have now to integrate, taking their
singular points specially into account.
1. The integral of an integer rational function.
When in is a positive integer, the power az is an analytic function
without exception in the entire plane. Therefore along any path of
integration:
z - n+l
S ds -- + C.
When the integration is extended to infinity, the integral function also
becomes infinite. The point infinity is a non-essential singular point.