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.

388

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

I. Ag,oz9 + A7,1 z7w + A5,2 5zw2 0,
II. As,2z5w2 + A1,5w5 ==,
III. Ai,5sw5 + Ao,7 W 7- 0.
To I. belongs the value:
9 —7 9 —5
~ --- 1 =2;
accordingly putting w = vz2, we obtain the quadratic for v:
Ag,o + A7,1v + A5,2 v2 == 0.
When this has unequal roots, v1 and v%, there are two expansions of
w by integer powers of S beginning respectively with the terms
w V1z2 and w = V2Z2.
These are not branched here. The next term in each is found by
substituting:
=v 2 ( + V3 z) and w= v2z (1 + v,)
and equating to zero the sum of the terms found to be of the tenth
dimension; thus:
A10,0, A10.0
'3 --- ~ V3 ^ - 'M
3 9 >A5,2 3vl' — A7,1 2 A5,2 v2- +A7,1
When the roots of the quadratic are equal, v1 == v = v, and thus:
A7,12 - 4A9,oA5,2 = 0 and 2A5,2 v + A7,1 = 0,
let us put w = (v -+ wV')Z2; in the equation between z and w' resulting
from division by z9, two values of w' vanish along with = 0. This
equation is:
A,,o + A7,1(v + w') + A5,2 (v + w')2 + A4,4 Z3 (v +- ')4 + A1,5 Z2 (v -- w')
+Ao,7z5(v+ '- ')7+..+Ao,oz=,
or, as it may be arranged in consequence of the condition for v:
Ao1,oZ + A5,2'2 +... 0.
Hence results:,=t.' ---.,
5,2
and consequently we obtain:
v -- v*2 + - 10 +.,
15,2
that is to say, a series that proceeds by powers of zo; two values of
the function ramify cyclically.
But when Alo,o = 0, the initial terms are: A,,5v5 + A5,2 w'2 0,
accordingly:
tS' + ] ' C',