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.

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Expansion of amlbiguous analytic functions. Bk. IV. ch. TTI.

But because j roots are to be equal to I in this equation, its first
j - i derivates with respect to A also vanish.
Now let the equation between z' and w' be investigated, as was
the original equation between s and w, by constructing the polygon
corresponding to the dimensions of its terms. Each side of the polygon
leads to a selection of terms of equal lowest dimensions, and each
simple root corresponding to this selection leads to the initial term
1 p'
0 = — q'
of an expansion, for which the denominator q' specifies the number
in the cycle. Now since
z l, and V=Q G q1+ IV) ',
the first two terms of the expansion of w in powers of z are:
1 p 1 p' —q'p,
I Pi 1 P'+ 7'P
wt=,q q +,Aq,'' q'q.
I 1
Here we can retain for each root Aqh and 'q' some one of its possible
values, while the roots of z assume all possible q q' values. Thus a
cycle of qlq' branches arises from this simple root; the next term in
the expansion is to be found by substituting:
1 p, 1 p'+-q'pl p'+q'p,+l
W _= qlzl + i'q'Z qlq' + V Z qlq..' etc..
But when there is a multiple root in the equation between w' and z',
let us similarly introduce the variables z" and w"; then as result of
substituting
I pI (1
W== AZq, I=- V', W -( Uq'+ w)P, =, — ( -W'+ )Z p,
provided this new substitution leads to a cycle between w" and z",
we shall have:
1 p"
wg --- 'q," q/,
therefore:
1 p' 1 p"+q"p'
c =-, q'q' -+- Z " 1" ' q q>
1 p, 1 p'-.q'p 1 p-qp' —p'q"_
V- =A q, zi+ q Z- ' ql + Aq-S qq,q"_ +.
a cycle of qlq'" values arises.
Now several such substitutions may be necessary, but ultimately
a finite number of them must lead to a simple root.
This we see as follows: Since 2 is a multiple root of the equation
of degree 71, j will be less than or at most equal to k1. Therefore there
is generally a decrease of multiplicity, and as the process is continued
we generally reach a simple point that presents a root also only