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.

~ 190. Classification of functions according to their singular points.

357

K, = — f(1) n-ll J Lin ff (- e-P )reif''pdp,:for r-=0,
round z' 0
or also
-= ~-J~ I + dz, round the point oo.
When there are no finite singular points, f(z) can be expressed by an
infinite series of ascending positive integer powers,
() G + + K A f- K2s- + K3X +.
which converges in the entire plane. In this case f(s) is said to be
an integer transcendental function.*) To this class the exponential
function belongs. The calculation of the coefficients K can moreover
then be reduced to any curve round the origin, and as no singular
points are included, we have:
KnJ - fX d7 = 1d fn (0
The following statements supplement the theorems resulting from
the expansion of the analytic function in a series of the Form VIII.:
An analytic function that has no singular point, and therefore
nowhere becomes infinite, is a constant.
It cannot be an integer function either rational or transcendental,
because either would have a singular point at infinity, non-essential
in the former case and essential in the latter.
Further: An analytic function without any essential singular point,
that nowhere vanishes, is a constant.
(The exponential function is zero at no finite point, its vanishing
point coincides with its essential singularity.)
An analytic function must assume every arbitrary value at least
once; otherwise it is a constant.
The value may possibly belong to the essential singular point.
An analytic function is determined when its values are given along
a finite portion of a curve however short.
For then all the successive derivates of the function at a point
can be calculated; therefore the series of powers in the neighbourhood
of this point is obtained, and this can be extended out beyond its
convergency (~ 186).
An analytic function is constant when it is constant along a finite
portion of a curve however short.
For then all its derived functions at a, point are zero.
*) The classification of transcendental analytic functions into integer and
fractional with the subdivision of these according to the number of their essential
singular points was given by Weierstrass in the Memoir cited in the foot-note
p. 130. (Also p. 148.) In it he also showed how the functions may be developed
analytically when the number of their non-essential singular points is unrestricted.