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.

~ 185. 186.

Taylor's series.

345

The series V. will generally not converge for every value t within
the region originally assumed, wherein f(z) was supposed analytic.
But we can vary the centre from a so as to arrive at a circle, and
with it at an expansion, that will include the required point. For, if
this point t be at a finite distance however small beyond the boundary
of the circle, let us draw from the point a to t any curve that keeps
always at a finite distance from the boundary curves. The circle
whose centre is a will meet this curve in a point a' between a and t.
The function f and arbitrarily many of its successive derivates can be
calculated from series V. for a point on the curve and arbitrarily near
a' within the circle; this point can then be made the centre of a new
expansion all whose coefficients are known. The point a" in which
the new circle of convergence crosses the curve is of course nearer
to t; and the continuation of this process must ultimately lead to a
circle that includes the point t, since the radii cannot become infinitely
small, because the path a a'a"... t is always at a finite distance from
the boundary curves.
This process of continuation can also be employed in the case of
an analytic function defined only by a series of powers, in order to
extend it out beyond its convergency into any domain not including a
singular point. (~ 87.)
Each such series of powers defines the function for a determinate
circular convergency and is called an element of the function.
According to the centre chosen for the expansion, different elements
of the function are obtained, moreover, one and the same value of the
argument belongs to different elements. But when the function is unique,
its different elements must lead to the same value for the same point.
When in this extension of the function by its elements we do not
pass out beyond some limited connected domain, the function exists
only for this domain.
Any pair of analytic functions of the complex' variable defined
arbitrarily within given domains are to be considered as belonging to the
same function, only, when the elements of one of the functions can be
derived from those of the other.
In this sense an analytic function is completely determined when
its value and the values of each of its derived functions at a point are
known, or in other words, when the values of the function are given
along a line however short. For then its derived functions can be
determined. Functions that are not thus connected, are to be regarded
as independent of each other.*)
*) Hankel: Untersuchungen fiber die unendlich oft oscillirenden und unstetigen
Functionen, p. 44 etc., 49 etc.; reprinted, Math. Annalen, XX, p. 104 etc.,
p. 109 etc.. Weierstrass, Monatsberichte der Berliner Akademie. 1870 August.