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.

362 General properties of unique analytic functions. Bk. IV. ch, II.
and the coefficients.ak are to be determined from the integral:
- f(z —) f' (z) Lt z
Now putting:
f(z) - f(oi) - F(-))
z —.a} 9(z- a)
whence:
1 (z- x)q'(z --.)
f'(z) = (Z - ( (Z - ( ))2-
we see that the integral breaks up into the following two parts:
i )(pJ -(- aL 21 J -- ( )k —l(z d-
The first part by ~ 185 is equal to
1(~l-((z — P )
I/O --- d - ' -
at the point z = a. The second part is equal to
1 d {-2 (q ( - cc))i-1 -'(z-)I ) 1 1 dk-P(Z - ))7 for
L-_ dZ7^-2 1 z^ —2 - Czk-1
so that:
I /(__ 11)'k1 (q(2 -- -c ))7 1 ilk-1 (q(z-C)V for a.
I ^ ^ -^)"-^^ -— ^ —d ----
ak — Ik-_ k2 for' a.
(10 - 1 k ) CZk-i = d -— 1
Accordingly we have the above solution stated in Lagrange's form:
From the equation: w'p(z) = z', in which cp signifies a unique
analytic function, the following expansion is found:,( '\ q 0 02g <(2 ())2 + 3 _ _' d 41 (
1=V (P1(02) + +,etc..
z' =-0 Z'=0 z'=0
Since:
f(.'\ -'f(A) " ~ i _A - ----
f(z) a)- /v/ _ u- co f'(a ) (z -- ax) - (z - c) '() - a)
the manner in which the preceding condition of convergence is brought
into relation with the function 9 (z') is as follows:
When r signifies the radius of the greatest circle round z'= 0 as
centre within which P(z')- -z''(') does not vanish, let us find the
smallest value assumed by w' -z upon this circle, this minimzum is
the radius of convergence for w'.
The domain of this circle in the plane w' can by inversion be
uniquely transformed into a domain of the plane z' included in the
circle of radius r, wherein also (p(z') is not zero.
Here the point w' = 0 corresponds to the point ' - 0.