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.

356 General properties of unique analytic functions. Bk. IV. ch. II.
When the function f(s) has a non-essential singular point at
infinity, and when moreover the number of its singular points in the entire
1
plane is finite, let us convert it by the substitution z - —, or
by z -- if the origin s = 0 be not singular, into a function of z'
that behaves regularly at infinity. For this function then, since the
new origin is now a non-essential singular point, the expansion:
C1 C2' C3' C()
t l(). -V 1 3 + i+
- -- Cl ) t' C -cl
(- t -cm) ( t' cC)
is valid in the entire plane, i. e.:
__ __ + (Z-_ + (AS )3 + * * * + 2 (d,)
f03() - _.+ +... j(- -
C m)
+ z - cm,) G 22 -'.
Hence follows: A function that has a non-essential singular point
at infinity, and a finite number of finite non-essential singular points,
is an improper fractional rational function. When it has no finite
singular point, it is an integer function.
Lastly, when the point infinity is an essential singular point, the
expansion:
K1 + 2 +.
t j- t'2 f in +
in the last series does not come to an end; therefore:
KIC, + AKt + K* -,z + *
also is an infinite series that converges for every finite value of z.
We have thus attained the most general form of the expansion:
When f(z) is an analytic function that has the non-essential or
essential singular points c, c2,. C.. in the finite plane, and also an
essential singular point at infinity, the expansion:
02' C' (W) C()
f(Z) + -- c+ + + 3 - Cl(
-c1 ( -ec)l (s - ce) 3 - CM2 (- — c,1)2
+ (z-%) + '+ * + KG z + KEZ9 + K3Za +...
+ (inCm)2 K
is valid for every finite value of a, and its coefficients C and K
are to be calculated by means of definite integrals, K in particular
from the form: