By I S Gradshteĭn; I M Ryzhik; Daniel Zwillinger; Victor Moll; Scripta Technica, inc
The 8th version of the vintage Gradshteyn and Ryzhik is an up-to-date thoroughly revised version of what's said universally through mathematical and utilized technological know-how clients because the key reference paintings about the integrals and distinctive capabilities. The e-book is valued through clients of prior variations of the paintings either for its accomplished insurance of integrals and detailed features, and in addition for its accuracy and worthy updates. because the first version, released in 1965, the mathematical content material of this e-book has considerably elevated end result of the addition of recent fabric, even though the scale of the e-book has remained virtually unchanged. the hot 8th version comprises totally new effects and amendments to the auxiliary stipulations that accompany integrals and at any place attainable such a lot entries include helpful 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 recent fabric no longer together with alterations to the limitations on effects that revise the diversity of validity of effects, which lend to nearly 35% of recent updates
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44011 b Then Let f (g(x)) and g(x) be continuous in [a, b]. Further, let g (x) exist and be continuous there. g(b) f [g(x)]g (x) dx = a f (u) du. g(a) Table of Integrals, Series, and Products. 00001-1 Copyright c 2015 Elsevier Inc. All rights reserved. 11 Power series ∞ q q(q − 1) 2 q(q − 1) . . (q − k + 1) k x + ··· + x + ··· = xk 2! k! k k=0 If q is neither a natural number nor zero, the series converges absolutely for |x| < 1 and diverges for |x| > 1. For x = 1, the series converges for q > −1 and diverges for q ≤ −1.
322 1. sinh2 x = 2. sinh3 x = 3. sinh4 x = 4. 323 1. cos2 x = 2. cos3 x = 3. cos4 x = 4. cos5 x = 5. cos6 x = 6. 324 1. cosh2 x = 2. cosh3 x = 3. cosh4 x = 4. cosh5 x = 5. cosh6 x = 6. 7 n n sin nx = n cosn−1 x sin x − cosn−3 x sin3 x + cosn−5 x sin5 x − . . ; 3 5 ⎧ ⎨ n − 2 n−3 2 cosn−3 x = sin x 2n−1 cosn−1 x − ⎩ 1 ⎫ ⎬ n − 4 n−7 n − 3 n−5 cosn−5 x − 2 cosn−7 x + . . 175) (n+1)/2 2. 12 n 2k − 1 sinh nx = sinh x k=1 (n−1)/2 (−1)k = sinh x k=0 3. sinh2k−2 x coshn−2k+1 x n − k − 1 n−2k−1 2 coshn−2k−1 x k n n cosn−2 x sin2 x + cosn−4 x sin4 x − .
12 2n+1 (−1)k k=0 n 4. 3) (2n − 1)! 15810 n 1. 2k 2n − k n−k − 2k+1 2n − k − 1 n−k−1 k = 4n − 2k 2n − k n−k − 2k+1 2n − k − 1 n−k−1 k 2 = 4n − 2k 2n − k n−k − 2k+1 2n − k − 1 n−k−1 k 3 = (6n + 13)4n − 18n 2k 2n − k n−k − 2k+1 2n − k − 1 n−k−1 k 4 = (32n2 + 104n) k=1 n 2. k=1 n 3. k=1 n 4. 15910 n 1. k=0 n 2. k=0 n 3. 16010 2n 1. k=n+1 n (−1)r 2. 211 uk = u1 + u2 + u3 + . . 212 |uk | = |u1 | + |u2 | + |u3 | + · · · , k=1 composed of the absolute values of its terms converges. 211 is said to converge conditionally.