Analysis

A Course of Modern Analysis: An Introduction to the General - download pdf or read online

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

ISBN-10: 110726684X

ISBN-13: 9781107266841

Filenote: PDF retail from ebsco. Please be aware this is often PDF retail however the writer has scanned and serarchable textual content or even hyperlinked the index quantity. I checked ebl, ebrary and ebsco, all have an analogous quality.
Publish 12 months note: First released in 1927
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This vintage textual content is understood to and utilized by millions of mathematicians and scholars of arithmetic thorughout the area. It provides an creation to the final concept of countless procedures and of analytic features including an account of the primary transcendental functions.

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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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Extra info 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)

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CO It is obvious that if, for all values of n greater than some fixed value r, | un+l | is greater than \un\, then the terms of the series do not tend to zero as n —+ x , and the series is therefore divergent. On the other hand, if is less than some number which is itself less than unity and independent of n (when n > r), we have shewn in § 2*36 that the series is absolutely convergent. The critical case is that in which, as n increases, - ^ un the value unity. In this case a further investigation is necessary.

The absolute convergence of this product depends on that of the series of which the (2m — l)th and (2m)th terms are z + m-rr But it is easy to verify that and so the absolute convergence of the series in question follows by comparison with the series i l l l i i i The infinite product in this last form is * therefore again absolutely ±— convergent, the adjunction of the factors e nn having changed the convergence from conditional to absolute. This result is a particular case of the first part of the factor theorem of Weierstrass (§ 7*6).

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).

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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) by E. T. Whittaker, G. N. Watson


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