By Daniel Zwillinger

The instruction manual covers, because it consistently has, numbers, geometry, trigonometry, calculus, specified services, numerical equipment, chance, and statistics.

New within the thirtieth version: communique idea, keep watch over idea, layout concept, Differential research, Graph thought, workforce thought, indispensable Equations, Markov Chains, Operations learn, Optimization innovations, Partial Differential Equations, Queuing idea, medical Computing, Tensor research, Wavelets, extra subject matters for someone and everybody Who makes use of arithmetic

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4! 6! (2n)! 2! cos x cos 2x cos 3x 2a 1 − 2 + 2 − 2 + ... (|x| < π). = sinh πa π 2a 2 a + 12 a + 22 a + 32 1 2 22n (22n − 1)B2n 2n−1 = x − x3 + x5 − · · · + x + ... (|x| < π/2), 3 15 (2n)! (Re x > 0), = 1 − 2e−2x + 2e−4x − 2e−6x + . . sinh x = x + sinh ax cosh x cosh ax tanh x = 2x 1 π 2 2 + x2 + 1 3π 2 2 + x2 + 1 5π 2 2 + x2 + ... 1 x x3 2x 5 22n B2n 2n−1 + − + + ··· + x + ... x 3 45 945 (2n)! = 1 + 2e−2x + 2e−4x + 2e−6x + . . coth x = 1 1 1 1 + + + ... +2x + 2 x π + x2 (2π)2 + x 2 (3π)2 + x 2 1 5 61 E2n 2n sech x = 1 − x 2 + x 4 − x 6 + · · · + x + ...

The bilinear transformation is defined for all z = −d/c. The bilinear transformation is conformal and maps circles and lines into circles and lines. −dw + b The inverse transformation is given by z = , which is also a bilinear cw − a transformation. Note that w = a/c. The cross ratio of four distinct complex numbers zk (for k = 1, 2, 3, 4) is given by (z1 − z2 )(z3 − z4 ) (z1 , z2 , z3 , z4 ) = . (z1 − z4 )(z3 − z2 ) The bilinear transformation is defined by w = If any of the zk is complex infinity, the cross ratio is redefined so that the quotient of the two terms on the right containing zk is equal to 1.

1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296, 340, 390, 448, 512, 585, 668, 760, 864, 982, 1113, 1260, 1426 Number of partitions of n into distinct parts, n ≥ 1 14. 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42 Euler totient function φ(n): count numbers ≤ n and prime to n, for n ≥ 1 15. 1, 1, 1, 0, 1, 1, 2, 2, 4, 5, 10, 14, 26, 42, 78, 132, 249, 445, 842, 1561, 2988, 5671, 10981, 21209, 41472, 81181, 160176, 316749, 629933, 1256070, 2515169, 5049816 Number of series-reduced trees with n unlabeled nodes, n ≥ 0 16.