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Elements of the Theory of Functions of a Complex Variable; With Especial Reference to the Methods of Riemann

Elements of the Theory of Functions of a Complex Variable; With Especial Reference to the Methods of Riemann


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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 edition. Excerpt: ...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 + /i (-a) + p3(z-a) + = i/(z), so that ip(z) represents a finite and continuous function for z = a, we obtain ri..a nm In this, by (9), 26, c"+, =-i-. fot(-a)" dz, the integral being taken along the circle C described round a. If we substitute in that integral z--a = r(cos $ + i sin 0), ----= id6, z--a we have =--f f(z)(z-a)"+id6. Now if, in order in the first place to consider the former case from this point of view, f(z) be infinite of the nth order for z = a, then (z--a)"f(z) is finite at a, and therefore (z--a)"+lf(z) is zero. Hence, if the radius of the circle C be made to tend towards zero, c'"+i) and with greater reason all succeeding coefficients, cM+2). c'"+3), vanish. Therefore the c( ) series contained in (4) ends with the term and changes (-a)" into the expression A, found previously under (1). If, on the contrary, the second possibility already mentioned occur, in which (z--a)'f(z) does not have a finite limit for any finite value of n, then none of the coefficients c(n) vanish, and the infinite series contained in (4) enters in place of the former expression A. In this case (z) is infinite of an infinitely high order for z = a, and at the same time, as remarked above, the discontinuity at a is of the second kind. Therefore the two kinds of discontinuity are also characterized by this, that in the first occurs an...


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Product Details
  • ISBN-13: 9781230097176
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 74
  • Spine Width: 4 mm
  • Width: 189 mm
  • ISBN-10: 1230097171
  • Publisher Date: 13 Sep 2013
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 150 gr


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