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Published online by Cambridge University Press: 20 January 2009
The functions f(t) and h(t) that occur in what follows are supposed to be integrable (L) in every finite interval in which they are defined; and the order of summability, which need not be an integer, is not negative.
page 84 note 1 See Titchmarsh, , op. cit., p. 37.Google Scholar
page 84 note 2 The corresponding theorem for cosine integrals is also true, and may be proved in the same way.
page 85 note 1 This theorem on integration by parts is due to G. H. Hardy in the case of integral r. See Cossar, J., Journal London Math. Soc., 16 (1941), 56.CrossRefGoogle Scholar
page 87 note 1 If, in (a, b), F is a bounded positive decreasing function and G (possibly complex) is integrable, then there is ξ (a ≦ ξ ≦ b) such that
.
page 90 note 1 The convergence of the expressions denoted by I 1 and I 2 will become apparent later. The same remark applies to several of the expressions In that follow.
page 90 note 2 If p = – 1, I 3 = 0.
page 92 note 1 SeeProc. London Math. Soc. (2), 48 (1944), 292–309.Google Scholar
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