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.

354

General properties of unique analytic functions. Bk. IV. cb. II.

From this expansion moreover we realise how the two kinds of
singular points differ in the nature of their expansions in powers.
The non-essential was characterised by the possibility of assigning
some positive integer mn, for which f(S) () - c)m at the point c is equal
to a finite quantity. Hence follows: When c is a non-essential point,
of infinitude m, all the coefficients higher than nC vanish for it. In
fact, because the radius Q can be chosen so small that f(z) (z - c)
shall differ inappreciably from a finite number G, the integral
/ f ( -..
(c)
will differ in value inappreciably from
G f( - c) d
(c)
and the value of this integral is zero. Conversely, if for a point c
all the higher coefficients C i,+l Cm,+2,.. vanish, it must evidently
be a non-essential singular point, because then ( - c)mf(s) remains
finite for S - c.
Every other point in which f(z) becomes infinite is an essential
singular point, and we now see that the essential singular point in any
analytic function behaves as it did in the exponential series where
we first took cognisance of it: there exists for its neighbourhood an
infinite series proceeding by negative powers.
The value of the reciprocal function -.-(- will be zero at every
non-essential singular point of f(z). For, when (- c)n2f(S) = G
for 2 ==~c, we have 1 — (-7- 0 for C e. But an essential
f(z) U
singular point remains such also for the reciprocal function. For if
in the region of an essential singular point the reciprocal function
1
[() were regular, the same should be the case also with f(z); and if
the singularity of f-) were only non-essential, f(s) should be regular.
Any essential singular point of f(s) is also essentially singular for the
function ( C denoting any arbitrary constant; we conclude
hence: Round the essential singular point a circle of radius Q can be
assigned, that certainly includes points at which f(2), /.(-).(Z) —
each becomes greater in amount than any arbitrarily prescribed number
K however great; i. e. points at which the amount of f(z) exceeds K, as
well as points at which it differs from zero, or from any number C, by less
than the arbitrarily small quantity -. In other words: The value of