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.

350 Expansion of analytic functions in series of powers.

lBk. IV. ch. II.

(1)1 W ----0
-,==0,
21 -a Ow2 +- 02 - - 0,
1 i IA 2 1 2 2 2
- Oul Wi + 2 2, W4 + 04 == 0
1 j ~ + 1 2 2 + 1 _w 1 2 2 a0
I-t + 3+ t 3 ) + W ~ W 9 + - - 0
L,1 awl i a2- a's
1n azui 19 + K 2u, + '1 ^3P2 1 _V 53. a, Lw. o. o,,,).
It is evident in this form also, that, for the regular point, in which
aqi - f a=:
does not vanish, all the differential coefficients as many as we may
form, continue finite.
Returning to the main result of the investigation of the algebraic
function in a regular point, we formulate it in the theorem:
Knowing a single value Wa' at a regular point =- a of an algebraic
function w given by f(zm, Wj) = 0, we can calculate the value of the
finction belonging to any other point S within the regular domain
round a, on expanding arbitrarily many terms of the series of powers
by a determinate succession of possible arithmetical operations.
We may frame the theorem still more completely: Knowing every
critical and singular point of an algebraic function given by f(zr, w)- 0,
and furthermore one of its values at a regular point, we can by a
determinate succession of finite arithmetical operations resolve the n-valued
finction by means of branching sections into n branches, each of which
is discontinuous only along the branching sections and becomes infinite
only in non-essential singular points; moreover, for any arbitrary point
every branch can have its value calculated by means of arbitrarily many
terms of an infinite series of ascending positive integer powers.
We arrive at this theorem by means of the method of continuous
extension of a function out beyond its circle of convergence repeatedly
described in the foregoing examples. Let the value of the function
given for a be denoted as belonging to the first leaf. From the point
a let us draw curves, each until it is arbitrarily near to one critical
point, but keeping at a finite distance from every other, and let us
surround that critical point by an arbitrarily small circle. Such a
curve along with the small circle is called a loop. By expansion we
can establish whether the circuit of the entire loop introduces a change
of the value of the function belonging to a; if it do not, the point