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.

Third Chapter.
Expansion of ambiguous analytic functions, specially of the algebraic
function.
193. Both the problem: of solving for tv the implicit algebraic
function defined by '(z, w) -- 0, and that just considered: of forming
the inverse function of a series of powers, led to ambiguous functions.
In each case we showed how Taylor's expansion is applicable within
restricted domains in which the function remains regular. The algebraic
function (~ 188) presented two kinds of points in which the function
w ceases to be regular: first, the points that were called non-essential
singular points, in which w becomes infinite, although its product by a
rational algebraic function remains finite; second, all the points that
were described as critical, in which the value of w counts as a multiple
root, for which therefore -afi' =- O. It may be, that both specialities
concur in the same point. In our last problem the ramifications formed
the irregular points. Generalising this problem to the inversion of any
unique analytic function in an arbitrary domain that also contains
essential and non-essential singular points, such singularities will also
occur in the inverse function.
Accordingly in what follows we shall investigate the properties
of ambiguous analytic functions in general, and specially those of the
algebraic function.
By an n-valued analytic function of a, defined for the entire plane
or for a finite domain, is meant a function which generally for each
value of S has n different values. Each of these values must satisfy
t f' r\f
the differential equation 0a +- i-fy = 0 - except in non-essential or
essential singular points. But furthermore the function must have
branching points. By a branching point or ramification is meant a
point S in which two at least of the corresponding values of the
function become equal, and in which moreover the values of the
function have changed when s has travelled round a curve enclosing
the point (~ 191). The ramification may happen to be also a singular
point. Now as we have already seen that in any domain that is