By Kenneth A. Ross, Charles R.B. Wright
Key gain: This ebook offers a legitimate mathematical therapy that raises easily in sophistication. Key themes: The ebook offers utility-grade discrete math instruments in order that any reader can comprehend them, use them, and circulation directly to extra complex mathematical themes. industry: A convenient reference for laptop scientists.
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Example 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,...