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. 1902 edition. Excerpt: ... coefficient of xm + r, (m + r--a) (m + r--b)Ar--(?n + r--i--c) (m + r--1--d) Ar-l = o. Differential Equation of the Hypergeomctric Series. 184. If in equation (1) of the preceding article we put a = o, and introduce a, /?, and y in place of b, c, and d by means of equations (4), we obtain.&(&-1 +y)y-x(# + a)( ...
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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. 1902 edition. Excerpt: ... coefficient of xm + r, (m + r--a) (m + r--b)Ar--(?n + r--i--c) (m + r--1--d) Ar-l = o. Differential Equation of the Hypergeomctric Series. 184. If in equation (1) of the preceding article we put a = o, and introduce a, /?, and y in place of b, c, and d by means of equations (4), we obtain.&(&-1 +y)y-x(# + a)(& + P)y = o, ... (1) or, since & = x--and &2 = x2---x--, in the ordinary no dx dx2 dx tation (1-)g+7-'++ 03 -# =. (2) This is, therefore, the differential equation of the hypergeometric series, -(a, 0, 7, x). Putting, also, a = o in the value of j we have y = AF(a, f3, y, x) + x-yF(a + 1-7, 0 + 1-7, 2-7, ) for the complete integral of equation (2). Since the complete integral of the standard form of the binomial equation of the second order, (1) Art. 183, is the product of this complete integral by x, it follows that the general binomial equation of the second order, equation (2), Art. 182, is reducible to the equation of the hypergeometric series in v and z by the transformations z = qx and y = zv. Integral Values of y and y'. 185. When a = b in equation (1), Art. 183, y = y' = 1, and the integrals y, and y2 become identical, so that there is but one integral in the form of a hypergeometric series. Again, if a and b differ by an integer, one of the series fails by reason of the occurrence of infinite coefficients. In this case, let a denote the greater of the two quantities, then y is an integer greater than unity, and y' is zero or a negative integer. The coefficient of x--1, in F(a, fi, y', x), is (- + i-y)... (a + tt-I-y) Q3+I-7)...(/?4---'-). (--i)!(2-y)(3-y)... (--y) This is the coefficient of x6--1, that is, of xa +-- in.y and isthe first which becomes infinite...
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Add this copy of A Treatise on Ordinary and Partial Differential to cart. $28.53, new condition, Sold by Revaluation Books rated 4.0 out of 5 stars, ships from Exeter, DEVON, UNITED KINGDOM, published 2001 by Adamant Media Corporation.
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