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.

342 Expansion of unique analytic functions in series of powers. Bk. IV. ch. II.

Not only do they present the simplest method of calculating any
function other than a rational algebraic function, but they define in
general all continuous functions of a complex variable whose first
derivate is determinate.
The way of arriving at this theorem was pointed out by Cauchy.
184. When f(s) is an analytic function without exception in a
simply connected domain inclusive of the boundary, and t -= + iv
denotes an internal point, the quotient t is in the same domain
likewise an analytic function, only that the one point s = t is a
non-essential singularity. An arbitrarily small circle drawn with radius
Q round t as centre will be the second
boundary curve of a domain in which
the above quotient is analytic without
exception. Integrating the quotient along
\ both the external boundary curve and
A (\S?~ ^this circle 9, keeping the ring surface
on the left, the sum of these two integrals
is zero. (~ 181.)
~/ G~_d We find the result of integrating
round the arbitrarily small circle of
Fig. 18. radius Q negatively, i. e. keeping its
interior on the right, as follows: For any point on the circumference
of this circle we have:
- t= Q(cosp+isinp)=-Qei', d Q - - ieiPd-p=Q-i(cosp+-isincp)dgp,
therefore:
y= 2t
J t -)- ds = - iJf(t + peiP)dp.
p=0
But since the function f is continuous in the neighbourhood of
the point z = t, we can choose 9 so small that the difference between
the values f(t) and f(t + 9ei~'), for all values of (p, shall be smaller
than a number 6 whose modulus is arbitrarily small. Accordingly
273 2zt
i f(t + - Qei)d(p differs from '~f(t)d(p 2inf(t)
0 0
by a quantity whose modulus is smaller than that of the arbitrarily
small number 2ztd; i. e. independently of the value Q we have the
integral:
(p- 2 7
J f (t + e (dI)d 2 equal to 2i tf(t).
1~ =-0

Therefore the following equality is established for the integral taken