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.

380

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

When one root of the equation k == 0 becomes infinite, this
means that fo,k = 0, and the degree of 0k reduces to k - 1. In order to
obtain that element of the function which belongs to the one infinite
root, let us first expand S as a function of w, as at the end of ~ 200.
The series begins with the term w2, or with some higher power of w
when further consecutive values of the derivates of z with respect
to w also vanish. When wsk' is the first term, we obtain by inversion
1
a k'-branched element of the function, having the initial term zk'.
The geometric statement of this case is: In the multiple point one
of the k branches of the curve has a tangent parallel to ordinates
having k' consecutive points on it, or having a contact of the order '- 1
with that branch of the curve.
The theorem still holds for each simple root of ck = 0 even
when there are multiple roots besides. But, for a multiple root it does
not hold; for, because vanishes for such a root, the values of the
higher derived functions for it are no longer generally finite. Thus
the question finally outstanding is: What is the form of the expansion
for a multiple value of w, finite or infinite?
202. Although this question can also be solved by successive
substitutions*), a process having the preference: that it employs only
the Theorem for the possibility of the expansion in a regular point
of an algebraic function in order to deduce from it the existence and
nature of the expansions in the singular point, still, since the general
investigations in this chapter establish the existence of the expansion,
it appears suitable that we should go back to the method given by
Newton, which has been elaborated by Puiseux.**)
Suppose the equation f(z + a, w + b) = 0, which defines the
function that is to be investigated in the neighbourhood of its
k-elementary point = 0, w = 0, arranged by powers of z and w.
Since we must henceforth assume that some partial derivates of the
kth order (specially fo,k) also may vanish, let us conceive the terms of
the equation arranged as follows. Take first the term independent of
w in which g occurs in the lowest power; there must be such a term,
for otherwise the factor w could be separated and the equation would
*) See Hamburger: Ueber die Entwickelung algebraischer Functionen in
Reihen. Zeitschrift f. Math. u. Physik, Vol. XVI. Nother: Ueber die singularen
Werthsysteme einer algebraischen Function. Math. Annal., Vol. IX, pp. 166-182.
N*) Newton in the letters to Oldenburg, June 13 and October 24, 1676.
Newton's method was explained and proved by Stirling, who says of it: "quae
est omnium quam quis excogitare potest, generalissima et elegantissima", Lineae
tertii ordinis, 1717; Cramer: Analyse des lignes courbes, 1750; Puiseux, see reference
in foot-note ~ 91, p. 154. Compare also the exposition given in Briot et Bouquet.