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.

344 Expansion of unique analytic functions in series of powers. Bk. IV. oh. 1I.
but it has also for each point within the domain higher derivates, and
these all, as many as may be formed, are analytic f[nctions.
186. Equation I. leads also to the expansion of the function f(t)
in an infinite series of ascending positive integer powers.
Let any point a be selected subject only to the condition: that
the greatest circle with a as centre which does not go outside the
boundary curve of the domain shall include the point t for which the
values of the function are to be calculated. Let this circle be adopted
as curve of integration of Equation I., then for every point S on its
circumference we have:
abs [t - a] < abs [z - a].
Accordingly
_L 1 = c - 1 +t t -- 6 + ta _ (t - a)2 - t a).
s —t — c a_ t-a Z- a l -- (s — a6)2 ^ '
2 —(
is a convergent series. By the Lemma proved in ~ 182, 3, therefore:
Ff() (t -d = J (12 + t - a) f( d )a) ~
/ vv y *2 z TT / i z - a ' v / J (< - a) +
+ (t - a)J- f (Z) ) da + (t-a)J- f d2..
and we obtain the equation:
f(t) = -, + (t i-) d)-:-! ( t
+ (t - CG) d + (t -- a)JI ( -- j l$i (1,.;
that can also in consequence of Equations I.-IV. be written in the
form:
V. f(t) = f(a) + (t - a)'(a) + - a) /"' (a) (t -- a) n (a).. + etc..
2 In
This series of powers certainly converges absolutely for every value
of t situated within the greatest circle round the centre a that does
not go outside of the boundary curves of the original domain. This
is Taylor's expansion for a complex function.
The contents of equation V. may be stated in the following words:
Given the value of a function and of each of its successive derivates
at a point a, if we know that the f/nction is analytic within a circle
round the centre a, the value of the function will be calculated for every
point within this circle by means of the infinite series of ascending
positive integer powers V.;
or: The value of a function and of each of its successive derivates
being given at a point a, it is an analytic function in the neighbourhood
of this point, only, when a circle of arbitrarily small finite radius can
be assigned round a as centre, within which the series V. is convergent.