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.

330

The definite integral of a unique analytic function. Bk. IV. ch. I.

the integral remains a unique function within a domain however
small round a, but this point itself may possibly be a singular point
for the integral.
A necessary condition in order that the integral up to the point
c may remain finite, in which case it is certainly continuous, is: that
for any number 6 however small, it shall be possible to assign a circle
with a as centre and with radius r, such that for all values of
z - = Q (cos + isin l)
for which Q < r, the amount of the integral
0
ff(a + Qei)eiPdQ
shall independently of i be smaller than 6. This will certainly be the
case ex. gr. when the function f(z) is so constituted that: for Q < r
an 0 < v <1,
abs [f(cx + Qe0d). c v]
is constantly smaller than a finite quantity G. For we have:
0 r r
abs J/'( + QCP)eidQ < abs [f( + ei)]dQ < G =Gand r can be chosen so as to make this quantity smaller than A.
We can also assign a condition which is sufficient in order that
the integral taken round the boundary of a point may vanish. This
condition is: that
Lim {( -- )f()} =0,
z = a
and therefore that a domain can be bounded off round the point a
within which the amount of the product (a - c)f(z) is equal to or
smaller than an arbitrarily small number 6.
For then, integrating round the circle with radius r, we have:
27z
abs ff(z)dz < abs [f(a + reiP) ire' dip < 2jr4,
and this value is arbitrarily small; it is therefore impossible that the
integral round the boundary should have a finite value, so that, since
its value is determinate and is the same for each boundary curve, it
must be zero.
We can now estimate the bearing upon integration of whatever
singularities there may be.
a) When the function f(z) loses the property of an analytic
function in a point by becoming discontinuous or by ceasing to satisfy
the equation while still remaining nite, this point
the equation + i - = 0O, while still remaining finite, this point
the quato ex ayf