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Elements of the theory of functions of a complex variable with especial reference to the methods of Reimann - Softcover
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1896 Excerpt: ... Consequently the occurrence of a polar discontinuity at a point a is always characterized by the property that the function becomes infinite of a finite order at that point. From this it follows at once that the case mentioned on p. 126 (note), that £(«) always becomes infinite at a point a for different paths of approach to this point, but infinite of different orders, is in fact not possible, but introduces a contradiction. In that case would receive the value zero for all paths of approach to a. But, as was shown above, j(z) becomes infinite of a definite order determined by that coefficient which is the first in (3) not to vanish. 1 Konigsberger, Vorlesungen ilberdie Theorie der ell. Funkt., I. S. 121. We now turn our attention to the second possibility mentioned on p. 133, namely, that there is no power of z--a with a finite, positive exponent /, for which the product (z--a,yj(z) acquires a finite value for all paths of approach to a. According to the preceding, this can occur only in the case of a discontinuity of the second kind. But the series derived (10), § 26, holds for the latter, because for that development the discontinuity occurring at a could be an entirely arbitrary one, the point a having been excluded by means of a small circle C. If in (10), § 26, we let Po+Pl (z--a)+ Pa(z-a)2 + = f(z), so that f/(z) represents a finite and continuous function for z = a, we obtain In this, by (9), § 26, c(n+1 = 1 r£(2)(2 _ a)'dz, 2 irt'J the integral being taken along the circle C described round a. If we substitute in that integral dz z--a = r(cos 6 + i sin 6), = id6, z--a we have c"+1 = J-f %(z)(z _ a)n+id0. Now if, in order in the first place to consider the former case from this point of view, £(z) be infinit...
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- PublisherRareBooksClub.com
- Publication date2012
- ISBN 10 1231698616
- ISBN 13 9781231698617
- BindingPaperback
- LanguageEnglish
- Number of pages74
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