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.

378

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

therefore:
1 2 3
w.. —b. (- a)Y -) +I ( )+- + ', =, f1 _
w b (-_ a )+, (cc)k + H ^-: +f AL fi,o
This critical point is accordingly a branching point of the
order k- 1; in it k of the n leaves required to exhibit the function
w uniquely are cyclically connected. When the point in question is
a real point of an algebraic curve with real coefficients, the real
figure of the curve has here a tangent with contact of the order
k - 1 as before, but parallel to ordinates. The curve also crosses the
tangent at the point (of inflexion) when Ac is an odd number.
201. The critical point has next to be investigated for the case
that fb/, and fi,o simultaneously vanish. We briefly denote henceforth
- a simply by z, and w - b by w, merely expressing thereby that
the origin of coordinates replaces the point S = a, w =- b; further
we shall write the value of - Wat that point = Wr. When all the
d zr
partial derived functions of the 2nd, 3rd,... (- 1)th orders vanish
besides the first two, the expanded equation is:
O=-f(z+-a,w+b)= o -{Ao A-, zk P-'+ f_~,pzkPP+-...-fo,i }
+i+1 { fk+l, 0ok +. } +-
In this case the point considered is called a k-elementary point*),
inasmuch as the system of values, =z 0 w = 0, along with the
several series of systems of values which satisfy f= 0 in its
neighbourhood, form k elements of the function defined by f = 0 at
that point. In fact putting - =0 in this equation, we obtain an equation
of the nth degree for w; assuming for the present that fO,k does not
vanish, k of its roots are zero. For, wk can be taken out as a
factor. Therefore lc leaves now meet in the critical point, and the
question arises whether they branch in it.
This requires us to investigate the quotient w: z near the
k-elementary point, in order to establish what values the first derived
function =c = Lim -, as well as the higher derived functions assume
in that point. Dividing by zk let us form the equation:
o = i I,o + "fk- 1,1 i + * * 1) fok-p,, ( ) + + fO,k (
j+ l f+l,0 + * * + )
which is of the nth degree in the quotient, but gives only lv finite values
of it when = 0 as roots of the equation:
N) o6ther, Math. Annal., Vol. IX, p. 169.