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.

328

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

180. If a new variable be introduced into the integral fJf()ds
by substituting z=;(z'), where (z') is an analytic function in the
entire domain within which the integral is to be formed, then by
~ 176 the property of integrability holds undisturbed with respect to
the new variable z', and thus:
ff(z) dz =jf ( (I') )'(z') dz
is an analytic function of a' with the derivate f((z'')). P'(z').
When the integral has to be extended over a domain reaching to
infinity, we can transform it by Inversion (~ 79) into an integral that
is to be investigated within a finite region. From the substitution:
IC/I dz'Z - aSie - P/1\dz-c1'
=- >- d = 2-, 2, we find: f(z)dz =- ) '
and to arbitrarily increasing values of s correspond values of A' with
arbitrarily small modulus. If then we have to integrate f f() dz
along a curve upon which z becomes infinite, we have:
Lim vf(s) dz = Lim -f f(+ ) z*'
forz= co for z =O
181. A number of corollaries depend on the Theorem of ~ 179:
1. When we form the integral along a closed curve that lies
within the simply connected domain, its value is sero.
For, when the path from s0 by z1 to z is one part of this curve,
and the path from zo by z2 to S the other, the integrals along these
paths are equal; and because along one and the same path:
Z 3 Z
- /(V)~ d ~ 178, 1),
the sum of the two integrals for the closed path from z0 to, and
from z to z0 is zero.
2. When the domain is not simply connected, both the integrals:
f ( iv) (dx + idy) =f (dx - vdy) +- i (vdx + udy)
formed for all the boundary curves of the domain in a positive circuit
are zero (~ 175,1.). If therefore we form f(z) dz in a positive circuit for
all the boundary curves of a multiply connected domain wherein f(z)
is an analytic ftnction, the value the integral assumes is zero.
Along each separate boundary curve it has a determinate value.