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.

The infinite series of powers is an analytic function.

147

Rn, = an+i (z + h)n+l + an+2 (z + h)n+2 + * * * + an+k (s + ht)nJk
shall be smaller than 6, as long as + -h lies in the convergency of
series 1), even when we put for each term its absolute value; when Y
is prescribed the value of n is determined. But now c can always be
chosen so great as to make the amount of:
z? ( ) h ' \ h2 (n)
n, k -- (n + Qn + + ( )
smaller than any quantity however small. For, each of the series )
converges absolutely; therefore in each of them can be found a place
c from which onwards the remainder of that series is constantly
smaller than a determinate quantity 6'. Hence we have:
mod 7~,k-n -'~() <( c + 6'. emoai [,
mod [Rn, - (Qn +. Qn + n. end[h
therefore we have also:
mod [e + - -n * (n)] < 28 + 6'. emod/].
This proves that the difference between the sums can be made arbitrarily
small by choice of n alone, so that series 2) and 5) must be identical.
Each point within the circle of convergence of 1) can therefore be
taken as the centre of an expansion, and its convergency will be at least
as great as the circle touching the inside of that original boundary circle.
Now from series 5) it follows that:
f(z + h) - f(z) 1,
- 72h,- = f, (z) + 2 f2 ( +, I' /'(z) + ~
therefore: Lim f(A + h) - f( (),
i. e. the first derived function of f(Z) is expressed by the infinite series 3):
f' () = fi(z) = a, + 2a2 +- 3a32z +. ncanz-1 +.* etc..
Or, the complex series of powers is differentiated by forming the series
of first derivates of its individual terms. This series converges within
the same circle of convergence as the original series.
Further, by differentiating the series for f' () it follows that:
f"(z) - f2 (z), similarly f"' (2) = f3 (), etc,.
Accordingly we have series 5):
fi( + JA) =- f(z) + 1fy () + n f " () t+ f ni(o ) + ~ ~
identical with Taylor's expansion for the function of a complex
variable given by an infinite series of powers.
Comparing this with 2) we find the meaning of the coefficients in
the expansion; we have:
ao-(= f(o) a, a, =f' (O), a2 ~ * - --- f L (0), etc.
10*