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.

~ 181.

The influence of singularities.

has no influence whatever upon the integral. In fact, for a positive
v either equal to or less than 1 we have.
Liam /( - a")f/'() } = 0.
The integral is still an analytic function in the point o, its derived
function is Lim f'() for a = a. (See moreover Remark c.)
b) But when the function f(z) becomes infinite at a point, whether
an essential or a non-essential singular point, this is always an infinity
point for the integral also. In fact, here
Lim {( - a)v (z) }
z =a
is not finite for v less than 1; because in a non-essential singular point
Lim {( --- c)"/'(,)
is finite only for vm equal to or greater than 1, but in an essential
singular point it is finite for no assignable m whatever. In the examples
of next Section we shall discuss this as well as the question whether
such a singular point is a branching point or not.
c) It is necessary also to consider whether possibly the function
f(s) may lose the analytic property all along a curve c situated
within the domain, by no longer satisfying the equation
a f a_
a+ -, =o
in any point of this curve, although continuous in its neighbourhood;
while the derived functions remain finite. This assumption however,
as we shall prove, involves a contradiction.
In fact if we surround the curve c by a boundary I arbitrarily
close to it, the integral round this boundary will vanish, because the
values of the function on the part of I to the right of c differ from
those to the left arbitrarily little when I closes in arbitrarily up to c;
therefore the sum of the integrals, being taken in opposite directions,
becomes arbitrarily small. Moreover the integral remains finite for
each point of the curve c. Therefore such a curve would not be
singular for the integral. Thence it follows that even in the points
of the curve we have:
U+ iV -(tu -- iv) (dx -f idy),
or the equations:
v, y x, y
U7=f (dX - vdy), V (vdx - ut dy).
Therefore the integral U + iV is an analytic function, whose derivate
is f(S) = u + iv. It will be shown subsequently, that the derivate
of an analytic function has also a derivate independent of the quotient