The Integration of Functions of a Single VariableUniversity Press, 1916 - 67 pàgines |
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a₁ absolute convergence algebraical function apply arithmetic arithmetic means ax²+2bx+c B₂ Bohr Bois-Reymond Borel Bromwich Cesaro's coefficients consider curve deduce defined denote differentiation divergent elementary function equation example exponential expressed factor finite number finite order follows formula G. H. Hardy generalisation given half-plane Hence increase of ƒ increasing function infinity of values integral function J₁ L-function of order Landau Lemma limit Lindelöf Lindelöf's Theorem log log logarithmic logarithmic means logarithmico-exponential Math mean value theorem method obtain ordinary Dirichlet's series polynomial positive number power series proof proved rational function result Riesz scales series is convergent series is summable sufficiently large values suppose tends to infinity Theorem 17 Theorem 41 Theorems 23 theory true typical means uniformly convergent variable write y₁ λη
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Pàgina 35 - We leave it as an exercise for the reader to show that the same ratio is found by considering an applied torque at some other co-ordinate.
Pàgina 30 - These are the cases of greatest interest ; and this course is one which we shall adopt in regard to a number of the theorems which follow. Further, it is easy to see that we may without loss of generality suppose C, the sum of the series, to be zero : this can always be secured by an alteration in the first term of the series. We are given that...
Pàgina 48 - Second edition, vol. 1, pp. 267-269. is pseudo-elliptic, and integrate it if it is, and there is reason to suppose that no such method can be given.
Pàgina 32 - ... is slower than that of any logarithm. 4.5. The importance of the logarithmico-exponential scales. We have seen that it is possible, in a variety of ways, to construct functions whose increase cannot be measured by any .//-function. It is none the less true that no one yet has succeeded in defining a mode of increase which is genuinely independent of all logarithmico-exponential modes.
Pàgina 75 - Fennicae, vol. 31, no. 2, 1902, pp. 1-48 (and Acta Math., vol. 28, 1904, pp. 37-64). F. Mertens (1) 'Über die Multiplicationsregel fiir zwei unendliche Reihen ', Journal fiir Math., vol. 79, 1875, pp. 182-184. G. Mittag-Leffler (1) 'Sur un nouveau théorème dans la théorie des séries de Dirichlet', Comptes Rendus, 22 Feb. 1915. CN Moore (1) 'On the summability of the double Fourier's series of discontinuous functions', Math.
Pàgina 2 - ... calculus integralis est methodus, ex data differentialium relatione inveniendi relationem ipsarum quantitatum ' (Institutiones calculi integralis, p.
Pàgina 65 - It is natural to ask whether anything can be said about the order of an when x is an algebraic number of higher degree.
Pàgina 32 - ... logarithmico-exponential scales. No function has yet presented itself in analysis the laws of whose increase, in so far as they can be stated at all, cannot be stated, so to say, in logarithmico-exponential terms.
Pàgina 48 - Aufsatze: über diejenigen Curven, deren Coordinaten sich als elliptische Functionen eines Parameters darstellen lassen (Borch.