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Proceedings of the Edinburgh Mathematical Society Volume 19-20

Proceedings of the Edinburgh Mathematical Society Volume 19-20


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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. 1901 Excerpt: ...is a decreasing function each of the differences fa-fa, fa-fa, is positive, and therefore, since each of the sums -!, s.2 is also positive, f(x) must be positive. Next let F(x) first vanish when x = xl and let X1 be in the interval (an-ll an). To findy) we have only to replace sn by zero; it is then obvious that f(xl) is positive since each of the sums s, s2, s _! is positive. Hence before F(x) vanishes there is a point at which the graph of F(x) passes below that of f(x). There is nothing in the proof to determine whether there are more points than one; if there are more points than one, let C1 be the first of them. Similar reasoning shows that if 1p(x) is negative in the first interval (a, e) the graph of J?(x) is at first below that of f(x), but must pass above it before F(jc) first vanishes. If we now transfer the origin of coordinates to C1, it is clear, by the same reasoning as before, that the graph of F(as) must cross that of f(x) before it again crosses the new a;-axis. If C2 is the first of these points of crossing, transfer the origin to C2 and proceed as before. Hence the most general form of the graphs of F(x) and f(x) will be as in Figure 3, and it is thus evident that if b = OM, there is always an ordinate F( ) or NR equal to MP or f(b) where = ON6. In the general case there may obviously be several values of; but there is always one such that a b. It may be observed that whether sr be positive or negative the equation f(x') = (&-&)-! + (fe-&). + + may be written in the form /(') = M- where sm is a quantity lying between the greatest and the least of the quantities s, s2, sn, and is therefore of the form (x)dx. J a But the form thus obtained for f(x') is not that ...


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Product Details
  • ISBN-13: 9781236376169
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 40
  • Spine Width: 2 mm
  • Width: 189 mm
  • ISBN-10: 1236376161
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 91 gr

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