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.

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

derived function even at the point a, and accordingly at that point
at' a '
the equation +- i =i- 0 must be satisfied.
In the unique analytic function, therefore, only essential and
non-essential singular points have to be considered, and in these all
the successive derived functions will be seen also to become infinite.
Let a circle described round the centre a with the radius X be
given. Suppose we know that a function f(s) is regular in the point
a and is analytic within the entire domain inclusive of the boundaries,
except at the points cl, c2.. c. within this circle which are to be
singular points, essential or non-essential. When a concentric circle
is described round a that excludes all the points c, a series can
be assigned within this smaller circle, ascending by positive integer
powers of s -- a, whose coefficients are the
derived functions at the point a. We are
/now, however, about to show that there also
exists an expansion valid for the entire
1./ A \ domain of radius R, no longer containing
x only positive integer powers of z with the
derived functions at the point a as their
coefficients, but enabling us to calculate
\ (i / 'the function for every point s within the
circle B, so that therefore there is no need
Fig. 19, of extensions of the series of powers, as in
the case of the previous expansion.
Let us enclose the points c1, c,... c. in circles of arbitrarily small
radius ); then within the one-leaved but multiply connected surface,
f(s) is an analytic function; and the sum of the integrals taken
round the circle a and round the circles c1, c2,.. c, keeping the surface
on the left, is zero. We can express this by the equation:
f() cu =Jf )d z + f (z) d + + (z) dz,
(a) (Cl) (C) (C n)
integrating now round each circle c so as to keep its area on the left.
When t signifies an arbitrary point within the multiply connected
surface, /() is in this surface an analytic function alike with f(z)
only that it has the point = t as a non-essential singular point.
Hence we have also:
J z t 2 f%-s (2 + 7ia- 2+ f.| 7 --.)
(a) 7 +J ) +... n.
( Z) ( t ) ( (Z C) ( Ze )

or: