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: ...when the arbitrary positive integers /jf, v are indefinitely increased; thus we have 1 JL r I aTM, n I rmsn + where is arbitrarily small, and hence, as desired, I t-A I am, n I rmsn Corollary, We have Herein r, s are arbitrary values such that the series converges for I x I = r-y = s an( M is any real positive quantity ...
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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: ...when the arbitrary positive integers /jf, v are indefinitely increased; thus we have 1 JL r I aTM, n I rmsn + where is arbitrarily small, and hence, as desired, I t-A I am, n I rmsn Corollary, We have Herein r, s are arbitrary values such that the series converges for I x I = r-y = s an( M is any real positive quantity greater, absolutely, than f(x, y), when x = r, y = s; so long as a001 0, we can always take p, a-so small that the inequality is satisfied. Thus if the origin is not a zero of the series it is an interior point of an assignable finite region within which no zeros are found. 51. Consider now the case when the origin is a vanishing point of the series. Arranged in powers of y let the series be A. + A.y-hA.y..., where A0) Alt A2y... are power-series in x of these Ad vanishes for x--0; we assume in the first instance that not all of A1 A2, ... vanish for #--0; let An be the first that does not, so that the series is of the form fix, y) = x(B0 + Biy+...+ Bn.lV) + (0 + xBn) + An+lyn +..., where B0, B1..., Bn_1 Bny An+lt... are power-series in x, and C is a nonvanishing constant. We shew now that a real positive quantity r can be assigned such that for any value of x less than r in absolute value, there are n values of y satisfying the equation /(, y) = o, all diminishing to zero with x, and that these are the roots of an equation yn+p1yn1+...+pn = 0, where Pi, p2-Pn ar power-series in x, vanishing for x--0, and converging for x r. Let A=A0, y) = Cp + lhf+..., and /i=/o-/(2/), so that f vanishes when x vanishes, identically in regard to y: Choosing r so that f0 does not. vanish for 0 y or, and so that f(x, y) converges for y a and sufficiently small x, and choosing ax so that 0 a1 a, we may Art. 50 implicit function theorem. 187 choo...
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Add this copy of Multiply Periodic Functions to cart. $61.07, good condition, Sold by Bonita rated 4.0 out of 5 stars, ships from Newport Coast, CA, UNITED STATES, published 2016 by Palala Press.
