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.

358 General properties of unique analytic functions. Bk. IV. ch. II.
191. Inversion of the unique analytic function.
When the analytic function f(z) — w has no singular point in a
finite domain, its inverse function s = t(w) is also analytic. For,
since there is a determinate derivate:
4 to
dw
there exists also a derived function of z with respect to w: namely
elz 1
dw~v/ f"(z) '
at each point within the region, and, putting w == + iv, we have. +,,i o.
This derivate of S with respect to w can become infinite only at
separate points at which f'(z) ==, and these will be found to be
branching points for the function = -(zv).
WVe may examine this in detail as follows, in order to assure
ourselves that the derived function -cdz depends uniquely on w. To
a determinate value z = — belongs a determinate value tw =-. On
inversion, a finite number of different values of S can correspond to
the single value w ===. Assuming then a determinate value for w
and considering one of the values s == a belonging to it, we may
enquire how S varies when the value of f is changed infinitesimally.
We shall show that this variation is continuous and unique, by
proving that the value of the derivate c- also is determinate as long
as w does not pass through a point at which the corresponding value
of z belongs to the equation '(Z) = 0, Again it is to be remarked,
that this equation possesses a finite number of solutions in any closed
domain; that to each solution belongs a determinate value of w, but
that inversely to each such value of w can belong different v'alues
of z of which however in general only one will satisfy the equation
f' = 0o.
The unique and continuous variation of S is perceived as follows.
Let a be a point for which f'(c) does not vanish. Then denoting a
neighbouring value of w by 3 + Aw, let the value a -- Az
correspond to it, thus when the right side of the equation
t 4+ AZv -=/'(t + AZ)
is arranged by powers of Ar, we have:
Az2 AA:
Aw A"(,I() + - 1 "() + - - /(") +. etc..
For- a given value of Aw within a finite domain for A,, different
values of A may possibly satisfy this equation; but since f'(a) is