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A Treatise Un Ordinary and Partial Differential Equations

A Treatise Un Ordinary and Partial Differential Equations


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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. 1889 Excerpt: ...hence derive the integral ys = (a, ft 1 + a + 0-y, I--). A comparison of this integral with yx or F(a, ft y, ) shows that from.any integral expressed by a hypergeometric series we can derive another integral of the same equation by making the above-mentioned change in the third element, and at the same time changing the fourth or variable element to what may be called its complement, that is, the result of subtracting it from unity. The process applies equally well to an integral of the form y =. wv, where v is a hypergeometric series; for a new integral of the equation for v gives a new integral of the equation for y. Thus the integrals y- yv and yA would lead to the three new integrals yv y6, and y8) which will be found in a subsequent article. XVI. CHANGE OF THE INDEPENDENT VARIABLE. 21$ 199. It is shown in Art. 194 that, when we change the independent variable to z =-, the binomial equation retains x its form. The equation considered in that article becomes that of the hypergeometric series when we put a = o, whence t =--a. Thus equation (1), Art. 194, gives the integral A comparison of this integral with F(a /?, y, x) gives a method by which we may pass from any integral in the form of a hypergeometric series to another in which the variable element is replaced by its reciprocal; and, as in the preceding article, the process applies also to an integral of the form y--wv, where v is a hypergeometric series. 200. If we start with the variable x, and alternately take the complement and the reciprocal, we obtain the following six values of the variable, x, 1 x, --, the seventh term of the series being identical with the first. The corresponding integrals derived from/, by the processes of Arts. 198 and 199, are y = F(a A y ) y5 r= F(, A 1 + a...


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Product Details
  • ISBN-13: 9781130733556
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 74
  • Spine Width: 4 mm
  • Width: 189 mm
  • ISBN-10: 1130733556
  • Publisher Date: 01 Mar 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 150 gr


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