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.

348 Expansion of analytic functions in series of powers. Bk. IV. ch. 1.
belonging to each is a multiple root. In the neighbourhood of any
other (regular) point each branch of w is an analytic function, and
consequently with any regular point a as centre must admit of an
expansion whose convergency extends at least as far as it can without
including any non-essential singula or critical point. Denoting by
wa' one of the values, necessarily simple, belonging to the point a,
the expansion is:
d. C\,d, to a) a)' d', i
+ (,Z; - a a, t'- a, -'a
Zw,- - \ + /- '2 " Zd,,++ ~~'^n + '",
where ( Ct ) denotes the value of the n^t derived function formed at
dzn
the point z = a, w - tWa'. We know a circle of convergence of
this series, when we have previously determined all such singular and
critical points of the function.
It has now become necessary to find the nth derived function.
We showed in ~ 94 how the first derivate of the implicit function
is calculated:
dw i _Ofe. __ f:
dz a(z ' 3w
and that it has a finite determinate value for every regular point
because at, is not zero. Every higher derived function is found from
this by successive differentiation: the partial differential coefficients
f, a/, alike with; are algebraic expressions in z and w; consequently,
as long as w is a unique analytic function of z, they are likewise
unique analytic functions of z. Therefore also their quotient is an
analytic function of z, and by the same rule of differentiation we find:
fa/' 2 t G d u f / _2 a dwx
d2'w + ~ V(vZ2 O'Wz dz) - aZ f z w v2 dZ~
The Theorem of the interchange of the order of differentiations holds
here, for, the algebraic functions _a_'-_, a2/ are continuous functions
of the variables z and w; therefore, inserting its value for?we have:
d2 f _ a t 2 f a 0 a 9 fa2 f \
-2l Zt J O a 2 O +V \OW J
C''d z-'! ' aGf 3zo
We have then in this expression on the right to substitute for z the
value a and for w the value v,'.