By E. T. Whittaker, G. N. Watson

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**Publish yr note:** First released in 1927

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This vintage textual content is understood to and utilized by hundreds of thousands of mathematicians and scholars of arithmetic thorughout the realm. It supplies an creation to the final thought of limitless methods and of analytic capabilities including an account of the main transcendental functions.

**Read Online or Download A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions (4th Edition) (Cambridge Mathematical Library) PDF**

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**Additional resources for A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions (4th Edition) (Cambridge Mathematical Library)**

**Example text**

We say that the series 2 vn consists of the terms of 2 un in a different order if a law 71 = 1 11 = 1 is given by which corresponding to each positive integer p we can find one (and only one) integer q and vice versa, and vq is taken equal to up. The result of this section was noticed by Dirichlet, Berliner Abh. (1837), p. 48, Journal de Math. iv. (1839), p. 397. See also Cauchy, RisumH analytiques (Turin, 1833), p. 57. 26 THE PROCESSES OF ANALYSIS [CHAP. II be an absolutely convergent series, and let S' be a series formed by the same terms in a different order.

L * Proc. London Math. Soc. (2) iv. (1907), pp. 247-265. These results, which are generalisations of Abel's theorem (§ 371, below), though well known, do not appear to have been published before 1907. From their resemblance to the tests of Dirichlet and Abel for convergence, Bromwioh proposes to call them Dirichlet's and Abel's tests respectively. 3'35, 3'4] CONTINUOUS FUNCTIONS AND UNIFORM CONVERGENCE [For we can choose w, independent of x, such that corollary, we have m+p 2 ano>n (or) n=m+l n=w»+l 51

I) A suitable interval is not to be bisected; for one of the parts into which it is divided might not be suitable. 54 THE PROCESSES OF ANALYSIS [CHAP. I l l This process of bisecting intervals which are not suitable either will terminate or it will not. If it does terminate, the theorem is proved, for CD will have been divided into suitable intervals. Suppose that the process does not terminate; and let an interval, which can be divided into suitable intervals by the process of bisection just described, be said to satisfy condition (B).