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The Differentiation of an Indefinite Integral

Published online by Cambridge University Press:  20 January 2009

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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 . 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 has such a derivate.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1924

References

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

cf. Narayan, Lakshmi, Bull. Calcutta Math. Soc., 8 (19161917) p. 71.Google Scholar

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