By Daniel Zwillinger
The guide covers, because it constantly has, numbers, geometry, trigonometry, calculus, unique features, numerical tools, likelihood, and statistics.
New within the thirtieth variation: verbal exchange thought, keep watch over concept, layout idea, Differential research, Graph conception, workforce idea, imperative Equations, Markov Chains, Operations examine, Optimization suggestions, Partial Differential Equations, Queuing conception, clinical Computing, Tensor research, Wavelets, extra subject matters for an individual and everybody Who makes use of arithmetic
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Extra info for CRC standard mathematical tables and formulae
Sample text
42 s7 (n) = 17 + 27 + 37 + · · · + n7 n2 (n + 1)2 (3n4 + 6n3 − n2 − 4n + 2). 24 s8 (n) = 18 + 28 + 38 + · · · + n8 n (n + 1)(2n + 1)(5n6 + 15n5 + 5n4 − 15n3 − n2 + 9n − 3). = 90 s9 (n) = 19 + 29 + 39 + · · · + n9 = n2 (n + 1)2 (2n6 + 6n5 + n4 − 8n3 + n2 + 6n − 3). 20 s10 (n) = 110 + 210 + 310 + · · · + n10 n = (n + 1)(2n + 1)(3n8 + 12n7 + 8n6 − 18n5 66 − 10n4 + 24n3 + 2n2 − 15n + 5). 13 NEGATIVE INTEGER POWERS Riemann’s zeta function is ζ (n) = α(n) = ∞ k=1 (−1)k+1 , kn β(n) = ∞ 1 k=1 k n . ∞ k=0 Related functions are (−1)k , (2k + 1)n γ (n) = ∞ k=0 1 .
Division: z1 z¯ 2 (x1 x2 + y1 y2 ) + i(x2 y1 − x1 y2 ) r1 z1 = = = ei(θ1 −θ2 ) . z2 z2 z¯ 2 r2 x22 + y22 |z1 | z1 = , z2 |z2 | arg z1 z2 = arg z1 − arg z2 = θ1 − θ2 . 3 POWERS AND ROOTS OF COMPLEX NUMBERS Powers: zn = r n einθ = r n (cos nθ + i sin nθ) DeMoivre’s Theorem. Roots: z1/n = r 1/n eiθ/n = r 1/n cos θ + 2kπ θ + 2kπ + i sin n n , k = 0, 1, 2, . . , n − 1. The principal root has −π < θ ≤ π and k = 0. 4 FUNCTIONS OF A COMPLEX VARIABLE A complex function w = f (z) = u(x, y) + iv(x, y) = |w|eiφ , where z = x + iy, associates one or more values of the complex dependent variable w with each value of the complex independent variable z for those values of z in a given domain.
X| > 1). x 3x 5x 7x (2n + 1)x 2n+1 1 E2n 1 (|x| < 1). = x − x3 + x5 + · · · + x 2n+1 + . . 6 24 (2n + 1)! 11 INFINITE PRODUCTS For the sequence of complex numbers {ak }, the infinite product is defined as ∞ k=1 (1+ ak ). A necessary condition for convergence is that limn→∞ an = 0. A necessary and sufficient condition for convergence is that ∞ k=1 log(1 + ak ) converges. Examples: • z! = ∞ k=1 • sin z = z 1 + k1 1 + kz z ∞ z 2k cos k=1 ∞ • sin πz = πz 1− k=1 • cos πz = ∞ 1− k=1 ©1996 CRC Press LLC z2 k2 z2 (k − 21 )2 ∞ • sin(a + z) = (sin a) 1+ k=0,±1,±2,...