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.

376

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

It is evident conversely: If the algebraic equation f(, w)== 0 be
such that we cannot pass from any one arbitrary initial value w by
arbitrary circuits of the branching points to every other value of the
root, the algebraic form must break up into rational factors; for, each
connected cycle satisfies an irreducible equation of lower degree with
rational coefficients.
200. Supposing now the irreducible equation f(zn%, wn) = 0 given;
it is required to establish criteria for estimating the properties of
each critical point and also a method of obtaining the expansions
valid in its neighbourhood.
In this investigation we may restrict ourselves to points in which the
values of z and w are finite; for, those points in which they are not
1 1
finite can be transformed by substituting = - and w = - respectively,
into points with finite values.
We shall first show how certain simple cases can be settled
without recourse to special methods. From this will emerge the most
general statement of the problem under any conditions whatever, and
its solution is presented in ~ 202.
Suppose the function w is known to assume the value b for a
determinate value z = a, then the algebraic form f(z, w) can be
expanded by powers of (a - a) and (w - b) (~ 94). The coefficients
of this expansion are the partial derived functions of f with respect
to z and w formed for the point = a, w= b; we shall briefly
denote them by:
( a f(z, v)) _ fp,.
k 0 ( p } \ k-pp
a,b
Then we have:
f(ze, wn) == fi,o(a - a) + fo,l((w - /)
+ i-jf2,o - a)2 + 21, (z - a) (w - b) + fo,2 (w- Zb)
+. + {i,o ( - a)' + kfk-l,i (- a)k- (W - b) +..
+ kVpfk-p,p( - a)k-P (w - b)P +.* * + f0,k(w - b)kj
a sum concluding with terms in which (z - a) rises to the mth and
(w - b) to the nth power.
When the system of values z= a, w =b, is a regular point
for the function w, fo, is not zero; in this case an expansion in a
series of ascending positive integer powers: