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By I S Gradshteĭn; I M Ryzhik; Daniel Zwillinger; Victor Moll; Scripta Technica, inc

ISBN-10: 0123849330

ISBN-13: 9780123849335

ISBN-10: 0123849349

ISBN-13: 9780123849342

The 8th version of the vintage Gradshteyn and Ryzhik is an up to date thoroughly revised version of what's said universally by means of mathematical and utilized technology clients because the key reference paintings about the integrals and specified services. The booklet is valued through clients of prior variants of the paintings either for its accomplished assurance of integrals and distinctive services, and likewise for its accuracy and beneficial updates. because the first variation, released in 1965, the mathematical content material of this e-book has considerably elevated as a result addition of latest fabric, even though the dimensions of the ebook has remained nearly unchanged. the recent 8th version comprises totally new effects and amendments to the auxiliary stipulations that accompany integrals and anyplace attainable so much entries include necessary references to their source.

  • Over 10, 000 mathematical entries
  • Most modern directory of integrals, sequence and items (special functions)
  • Provides accuracy and potency in work
  • 25% of latest fabric now not together with alterations to the limitations on effects that revise the variety of validity of effects, which lend to nearly 35% of recent updates

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Extra info for Table of integrals, series, and products

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211 diverges. 211 converges absolutely. 211 diverges. 224 Suppose that f (x) is a positive decreasing function and that ek f ek =q k→∞ f (k) ∞ k=1 f (k) converges. If q > 1, this series diverges. (Ermakov) lim for natural k. 225 Suppose that uk |vk | q =1+ + p , uk+1 k k where p > 1 and the |vk | are bounded, that is, the |vk | are all less than some M , which is independent of k. 211 converges absolutely. If q ≤ 1, this series diverges. 226 Suppose that a function f (x) defined for x ≥ q ≥ 1 is continuous, positive, and decreasing.

1 k=1 n (n + k)! (n − k)! 13112 k=1 1 1 = C + ln n + − k 2n ∞ k=2 Ak , n(n + 1) . . (n + k − 1) JO (59), AD (1876) where 1 1 x(1 − x)(2 − x)(3 − x) · · · (k − 1 − x) dx k 0 1 1 , A3 = , A2 = 12 12 19 9 A4 = , A5 = , 120 20 863 1375 , A7 = A6 = 504 168 Ak = n 2 3 − 1 B4 1 1 B2 = (C + ln n) + ln 2 + 2 + + ... 14 Sums of products of reciprocals of natural numbers n 1. k=1 n 2. k=1 n 3. k=1 1 n = [p + (k − 1)q](p + kq) p(p + nq) GI III (64)a 1 n(2p + nq + q) = [p + (k − 1)q](p + kq)[p + (k + 1)q] 2p(p + q)(p + nq)[p + (n + 1)q] GI III (65)a 1 [p + (k − 1)q](p + kq) .

0 < θ < 1] , (Schl¨ omilch) where ψ(x) is an arbitrary function satisfying the following two conditions: (1) It and its derivative ψ (x) are continuous in the interval (0, x − ξ); and (2) the derivative ψ (x) does not change sign in that interval. If we set ψ(x) = xp+1 , we obtain the following form for the remainder: Rn (x) = (x − ξ)n+1 (1 − θ)n−p−1 (n+1) f (ξ + θ(x − ξ)) (p + 1)n! Rn (x) = 6. [0 < θ < 1] Rn (x) = x 1 n! 12 ∞ f (x) = k=0 xk (k) x x2 f (a) = f (a) + f (a) + f (a) + . . k! 1! 2!

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Table of integrals, series, and products by I S Gradshteĭn; I M Ryzhik; Daniel Zwillinger; Victor Moll; Scripta Technica, inc

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