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The Theory of Functions of a Real Variable and the Theory of Fourier's Series

The Theory of Functions of a Real Variable and the Theory of Fourier's Series


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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. 1907 Excerpt: ... this condition is satisfied, the value of $-%-is infinite with definite sign. Oxq oy0 239. The differential coefficient----is the repeated limit oy" ox0 lim lim /fa ] h, y0 4-A)-/(g" + A, y)-/(a;, y" + A;) +/($, y.). 4=0 A=0 WK 9s/" df and thus the conditions that the relation--ir-= --holds are identical dxody, dy0dx" with the conditions that the two repeated limits may be identical. The necessary and sufficient conditions may be accordingly obtained by applying the conditions contained in either of the theorems in 233, and 235, to the function F(h, k) r/"+ ' y" + S "/(+ h' Vl z/faj Vl+ Q +/( V). fife It is however convenient, for application in particular cases, to have sufficient conditions relating to the partial differential coefficients in the neighbourhood of the point (x0, y0). The following theorem will be established: --d'fix, v). If (1), "L.' y exist and be finite at all points in a two-dimensional neighbourhood of the point (x0, y0), except that its existence at (a?0 y0) is not d'fix v) assumed, and (2) the point (x0, y0) be a point of continuity of J '" with respect to (x, y), the limit of this partial differential coefficient at x," y0) being a definite number A, and (3) f(x, y) be continuous with respect to x at (x0, y0), then V, -. V both exist, and have the same value A. 8y0Sa;0 dx0dy0 It will be observed that the condition (1) implies the existence of ' at all points in a neighbourhood of (x0, y0), except at that point ox itself, and that it is continuous with respect to y. From the condition (2), we have, corresponding to an arbitrarily chosen positive number e, /( + + =il+a(M); where a e, provided h, k are each less than some f...


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Product Details
  • ISBN-13: 9781153180405
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 270
  • Spine Width: 14 mm
  • Width: 189 mm
  • ISBN-10: 1153180405
  • Publisher Date: 22 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 485 gr


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