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Difficult questions arose Post Date: Sun, 27 Jul 2008 16:38:12 +0000 Those of the third kind rest mainly on the consideration of the limit of the difference either of consecu tive terms or of their reciprocals. In the generalised criteria of the second kind he does not consider the ratio of two con secutive terms, but the ratio of any two terms however far apart, and deduces, among others, two criteria previously given by Kohn and Ermakoff respectively. Difficult questions arose in the study of Fourier s series. Autor of the post: Undefined Dirichlet made the first thorough Post Date: Sun, 27 Jul 2008 16:20:00 +0000 79 Cauchy was the first who felt the necessity of inquiring into its convergence. But his mode of proceeding was found by Dirichlet to be unsatisfactory. Dirichlet made the first thorough researches on this subject (Crelle, Vol IV). Autor of the post: Undefined Dirichlet s conditions are sufficient Post Date: Sun, 27 Jul 2008 16:03:10 +0000 They culminate in the result that whenever the function does not become infinite, does not have an infinite number of dis continuities, and does not possess an infinite number of maxima and minima, then Fourier s series converges toward the value of that function at all places, except points of discontinuity, and there it converges toward the mean of the two boundary values. Schlafli of Bern and Du Bois- Eeymond expressed doubts as to the correctness of the mean value, which were, however, not well founded. Dirichlet s conditions are sufficient, but not necessary. Autor of the post: Undefined Bdernann inquired what properties Post Date: Sun, 27 Jul 2008 15:48:03 +0000 Lipschitz, of Bonn, proved that Fourier s series still represents the func tion when the number of discontinuities is infinite, and established a condition on which, it represents a function having an infinite number of maxima and minima. Dirich- let s belief that all continuous functions can be represented by Fourier s series at all points was shared by Eiemann and Hankel, but was proved to be false by Du Bois-Keymond and Schwarz. Bdernann inquired what properties a function must have, so that there may be a trigonometric series which, whenever it is convergent, converges toward the value of the function. Autor of the post: Undefined Eiemann rejected Cauchy s defini Post Date: Sun, 27 Jul 2008 15:35:43 +0000 He found necessary and sufficient conditions for this. They do not decide, however, whether such a series actually repre sents the function or not. Eiemann rejected Cauchy s defini tion of a definite integral on account of its arbitrariness, gave a new definition, and then inquired when a function has an integral. Autor of the post: Undefined Doubts on some Post Date: Sun, 27 Jul 2008 15:20:42 +0000 His researches brought to light the fact that con tinuous functions need not always have a differential coeffi cient. But this property, which was shown by Weierstrass to belong to large classes of functions, was not found necessarily to exclude them from being represented by Fourier s series. Doubts on some of the conclusions about Fourier s series were thrown by the observation, made by Weierstrass, that the integral of an infinite series can be shown to be equal to the sum of the integrals of the separate terms only when the series converges uniformly within the region in question. Autor of the post: Undefined This was done by Heinrich Post Date: Sun, 27 Jul 2008 15:07:34 +0000 The sub ject of uniform convergence was investigated by Philipp Lud- wig Seidel (1848) and Gr Stokes (1847), and has assumed great importance in Weierstrass theory of functions. It became necessary to prove that a trigonometric series repre senting a continuous function converges uniformly. This was done by Heinrich Eduard Heine (1821-1881), of Halle. Autor of the post: Undefined Improve ments and sirnplications Post Date: Sun, 27 Jul 2008 14:51:16 +0000 Later researches on Fourier s series were made by Cantor and Du Bois-Beymond. As compared with the vast development of other mathe- matical branclies ; the theory of probability has made very insignificant progress since the time of Laplace. Improve ments and sirnplications in the mode of exposition have been made by De Morgan, Boole, Meyer (edited by Czuber), Bertrand. Autor of the post: Undefined Worthy of note Post Date: Sun, 27 Jul 2008 14:36:10 +0000 Cournot s and Westergaard s treatment of insurance and the theory of life-tables are classical. Appli cations of the calculus to statistics have been made by Quetelet (1796-1874), director of the observatory at Brussels; by Lexis ; Harald Westergaard, of Copenhagen ; and Dusing. Worthy of note is the rejection of inverse probability by the best authorities of our time. Autor of the post: Undefined For example, if a man Post Date: Sun, 27 Jul 2008 14:21:07 +0000 This branch of probability had been worked out by Thomas Bayes (died 1761) and by Laplace (Bk II, Ch VI of his TMorie Analytique). By it some logicians have explained induction. For example, if a man, who has never heard of the tides, were to go to the shore of the Atlantic Ocean and witness on m successive days the rise of the sea, then, says Quetelet, he would be entitled to conclude that there was a probability equal to that the sea would rise next day. 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