Proceedings of the Edinburgh Mathematical Society

Table of Contents - Volume 43

Research Article

The Differentiation of an Indefinite Integral

J. M. Whittaker

Theorem 1 needs very little explanation. It is the converse of the well known theorem that the indefinite integral F(x) of a function f(x) possesses a derivate on the right at every point at which f(x + 0) exists. If f(x + 0) does not exist, nothing can be said as to the existence or otherwise of F+(x); but in a general way we might expect that the integral of a function which oscillates comparatively slowly, say sin (log x) at x = 0, would be more likely to possess a derivate than that of a function which oscillates more rapidly, say S0013091500036385_inline1. It appears from Theorem 1 that this is not by any means the case. In fact the integral of sin (log x) has not a definite derivate at x = 0 while that of S0013091500036385_inline1 has such a derivate.

(Received May 15 1924)

(Revised June 05 1924)

Notes

* Hobson. Functions of a Real Variable. 2nd Edition, I. p. 464. [Google Scholar]

† cf. Narayan, Lakshmi, Bull. Calcutta Math. Soc., 8 (1916–1917) p. 71. [OpenURL Query Data]  [Google Scholar]

‡ Cf. Whittaker and Watson. Modern Analysis, 3rd Edition, p. 156. [Google Scholar]