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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. 1906 edition. Excerpt: ...is readily obtainable. Noting namely that we have Gt = t, -')mFt, it suffices to multiply the 3rd, 4th, ... nth. rows of the determinant in (12) by tm, 52m, ... - ' respectively, dividing at the same time the 2nd, 3rd, ... (n--1)th columns by 4m, 42 ', ... S -2' ' respectively, in order to obtain the relation (19) 5(M)= Vn-1)mB(z, v). Field, . 12 This relation could also be immediately derived from the properties which define the functions R(z, v) and R(l, 7j). Namely the property that the coefficients of t2n-2, ...v in the product F (z, v). R (z, v) must vanish, and the property that the coefficients of r/ -2, ...t in the product G(i, y).Ri, fi)imnFz, v).Ri, ri) must vanish, are one and the same, and, since this property determines each of the functions R (z, v) and R(t, i) to a factor in t, it follows that these two functions can only differ by such a factor. From the further property that the coefficient of t -l in R(z, v) and the coefficient of V-1 in R(i, ti) must be unity, it follows that the latter function is obtained on multiplying the former function by the factor CHAPTER X. Rational functions of unrestricted character for z = 00. Form of the general rational function of (z, v) which becomes infinite only for the value z = x, . General form of a rational function of (z, v) which, in addition to infinities at oo, may possess an assigned set of infinities corresponding to finite values of the variable z. We shall now consider the form of a rational function of (z, v) as related to the values of the variables for which it becomes infinite. We have seen that any rational function of (z, v) can be reduced to the form where the coefficients h are rational functions of z. By application of the principle o...

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