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.

384

Expansion of ambiguous analytic functions. Bk. IV. ch. III.

and from this for determining the coefficient v in the expansion
pi
w —=v.zq we find the equation of the degree h - P-i:
A,_i-ll_- + -AA,-vs — l + Ao,, v-~i-l 0 -The result is therefore stated: When the polygon consists of i sides,
p
i different initial terms of expansions w = v. zq are possible; and if
the coordinates of the vertices of the polygon are called:
1, ~;, fI1; "2,,2;.* * a i-, pi-; 0, h,
the numbers of such possible expansions belonging to the values:
l - a P2 a_ - 02 Pi-1 ai- -2 i- -1.i di -1
qt P1 q2 P 2- P1 qi-1 pi -1 pi-2;qi h - i-tare respectively:
1; 2 1 - 1.. i-1 - Pi-2; h - O-1,
on the whole therefore h expansions of the h values of w that vanish
for z = O have been proved possible.
The quotients p: q1, P2: q,...p: q form a decreasing series
of numbers, as a glance at the figure shows, because the tangents of
the angles between the sides of the polygon and the positive axis of
x are the negative values of these numbers.
It must also be shown that these expansions are all actually
necessary in order to obtain the h expansions of w.
Let us consider the /1 possible expansions belonging to the first
side of the polygon, and so to the ratio p,: q; this number /3 is
either equal to q1 or is a multiple of it, suppose /B = k q1. Now since
for each point a, j/ that lies upon this side of the polygon,
a + 3 == a + F q1 = 1,
we have:
I — a p,
therefore also each such f3 is equal to q1 or to some multiple of q,,
suppose =- kq1.
In the equation from which the corresponding value of v is
calculated let us substitute Aql for v, thus it becomes:
A,0 + -.2Ac, ~ -k + All k1 = 0.
This presents k, finite values for A, some of them moreover may be
equal. Then to each simple root of such an equation corresponds a
cycle of q, values; the expansion begins with the term:
1 pt
W - am. = 1,