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.

~ 189. 190.

Essential and non-essential singular points,

355

a function at any essential singular point is completely indeterminate
between infinite limits, in the vicinity of the point it converges to
every arbitrary value. In the case of the exponential function this
property was already pointed out. ~ 82, 4.
The integral of f(z) taken round the arbitrarily close boundary
of an essential or non-essential singular point c is zero, only when,
in the expansion relative to the point c, the coefficient of the term
-1 is zero. The integral taken up to the singular point always
z- c
becomes infinite; its infinitude is m - 1 for a non-essential point of
f(s) of the order m, and for a non-essential point of the order 1
the integral becomes logarithmically infinite.
190. We have now to examine how the domain of f(s) admits of
extension out beyond the domain R: Let R' > R be the radius of a
circle round the centre o, in it let f(z) be generally an analytic
function, only with the additional singular points +1,... C+k;
then in Expansion VIII. the coefficients Ao A,,A2... change,
while all the coefficients C remain as they were, and new terms arise
by the points Cn+,. n+k-.
Assuming first that the function f(s) is not singular at infinity,
but converges for = —oo to a determinate finite value G, and that it
has in all cases a finite number in of singular points in the entire
plane; then the integrals in Series VI. that determine the coefficients
A all converge to zero, for R can be chosen so great that f(s) shall
differ inappreciably from the finite number G, accordingly we have also:
2 n
modJ {f(z <J Jmo CI- d(p
0
arbitrarily small. Only, the value of the first term:
A i f1 t'I-1
A0 = fiJl c a
is equal to G.
Therefore a unique analytic function which has in singular points
in the entire plane and behaves regularly at infinity is of the form:
(t) _c_ c( 02 (m) (+) G
tf(t -c (t-c- ) (tc) t- c (t-C-, (t- - cG.
A function which has only finite non-essential singular points and
is regular at infinity, differs from a proper fractional rational function
only by an additive constant. For then the terms that refer to the
various points c all come to an end, and when we reduce them to
a common denominator, the order of the numerator is at least one
less than that of the denominator.
23*