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.

324

The definite integral of a unique analytic function.

B31c. I1V. ch. I.

1) f )d = ff (),
Zo Z
only that both integrals must refer to the same curve. Again:
Z Zk Z
2) Sf(z) dS =gff() cdl +Jf(s) d,
Zo 0o Zk
only that the paths of integration from z0 by Zk to Z or from go by Z
to Zk must be the same in all the integrals.
There is likewise here a Theorem of the Mean Value; although
it has not as simple a form as for the real integral.
In fact, from the equation:
z T T
f(z)d -Sfi (t) { 9'(t) + i '(t) d t- iJf2 (t) { '(V) )+ i V(t) l,
Z0 to to
follows:
z
f f()dz = M(Z - o) + iMj(Z - o),
0o
where Ml/ signifies a mean value of f (t) and J12 a mean value of f (t).
When these functions are continuous along the path of integration, this
equation can be given the form:
z
3) ff,()cs I= {f, (to + 0 (T - to)) + it (to + (T- to)) } (Z 2).
Zo
The values of 0 and 0', however, will in general be different.
It can further be deduced from this equation, that the integral
in every point of the path of integration presents a function of the
complex variable, and moreover generally an analytic function. For if
2 denote an arbitrary point on the path of integration, the differential
quotient in an arbitrary direction of the integral:
Zo
with respect to its upper limit is derived by forming the quotient:
Z+71
z
where h signifies any quantity converging to zero and the integration
is along any path between the points and - Ai. If the values t
and t -+ k correspond to these points in such a way that when h = (0