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.

258 The definite integral as the limiting value of a sum. Bk. III. ch. VI.

For, if f(x) be a function that increases as c is approached, but in
such a way, that in the neighbourhood of the point c, for x < c:
abs f(x) is < A
(C - X),
A being any finite quantity, and v a positive proper fraction; then:
C-E
jf(x)dx will be smaller than A f- x =- A (-v 1 —
C J C-X)Y 1 — v
c-a c-a
and as long as 1 -v > 0 this expression converges to zero along
with a and e. Even when the order of becoming infinite (infinitude)
differs from unity by no assignable number, when ex. gr.*)
A 1
abs f(x) < - g — ))
c-x (log (c - x))1 '
the quantity:
Jxdx <A ) (lo ( ))V= (log ())- - (log (a))-"
J t'(x) ~lx < A (C ~- x) Gog (C - x))' + ~ —
c - a c - a
converges to zero along with a, provided v is positive.
But the above condition for a determinate limiting value of the
definite integral is certainly not fulfilled, when the function that is to
be integrated becomes determinately infinite in the first or in a higher
order; i.e. when its infinitude > 1. For, if:
abs f(x) > - A
~C- ~ ~E^ C — S -
(x) x is > A - = A (log a - log a),
c- a c-a
then as a converges to zero, these logarithms become infinite and
their difference is completely indeterminate.**)
In case of a function that becomes indeterminately infinite at a
point, the condition is satisfied without requiring any restriction as to
the order of becoming infinite; thus ex. gr. the function:
*) Riemann: ges. Werke, p. 229. See a general remark on the universality
of logarithmic criteria by Du Bois-Reymond, Journal f. Math., Vol. 76, p. 88.
**) log a - log =- log (a: s) assumes arbitrary values, according to the
way in which the ratio of the vanishing quantities a: s is determined. Making
a =, gives a value for which the logarithm vanishes. We could therefore in
this sense speak of a finite value of the integral f -, that results from a
determinate way of approaching the infinity point. Such special determinations,
which were frequently employed by Cauchy, are styled singular integrals,
but, as Riemann noticed, they have not been adopted in framing the general conception, because they require special arbitrary investigations in each calculation.