Appendix A Mathematical Tables and Algorithms This appendix provides lists with some definitions, integrals, trigonometric relations and the like. It is not exhaustive, presenting only that information used throughout the book and a few more just for complementation purposes. The reader can find extensive tables available in the literature. We recommend the works by Abramowitz and Stegun [1], Bronshtein, Semendyayev, Musiol and Muehlig [2], Gradshteyn and Ryzhik [3], Polyanin and Manzhirov [4] and Zwillinger [7]. We also recommend the book by Poularikas [5], though its major application is in the signal processing area. Nevertheless, it complements the tables of the other recommended books. In terms of online resources, the Digital Library of Mathematical Functions (DLMF) is intended to be a reference data for special functions and their applications [6]. It is also intended to be an update of [1]. A.1 Trigonometric Relations ∞ x3 x5 x7 (−1)n x2n+1 sin x = x − + − +···= (A.1) 3! 5! 7! (2n + 1)! n=0 ∞ x2 x4 x6 (−1)n x2n cos x = 1 − + − +···= (A.2) 2! 4! 6! (2n)! n=0 1 sin2 θ = (1 − cos 2θ)(A.3) 2 1 cos2 θ = (1 + cos 2θ)(A.4) 2 1 sin2 θ cos2 θ = (1 − cos 4θ)(A.5) 8 1 cos θ cos ϕ = [cos(θ − ϕ) + cos(θ + ϕ)] (A.6) 2 1 sin θ sin ϕ = [cos(θ − ϕ) − cos(θ + ϕ)] (A.7) 2 D.A. Guimaraes,˜ Digital Transmission, Signals and Communication Technology, 841 DOI 10.1007/978-3-642-01359-1, C Springer-Verlag Berlin Heidelberg 2009 842 A Mathematical Tables and Algorithms 1 sin θ cos ϕ = [sin(θ + ϕ) + sin(θ − ϕ)] (A.8) 2 sin(θ ± ϕ) = sin θ cos ϕ ± cos θ sin ϕ (A.9) cos(θ ± ϕ) = cos θ cos ϕ ∓ sin θ sin ϕ (A.10) tan θ ± tan ϕ tan(θ ± ϕ) = (A.11) 1 ∓ tan θ tan ϕ sin 2θ = 2sinθ cos θ (A.12) cos 2θ = cos2 θ − sin2 θ (A.13) 1 sin θ = (eiθ − e−iθ ) (A.14) 2i 1 cos θ = (eiθ + e−iθ ). (A.15) 2 A.2 Gaussian Error Function and Gaussian Q-Function Error function: x 2 2 erf(x) = √ e−t dt. (A.16) π 0 Complementary error function: ∞ 2 2 erfc(x) = 1 − erf(x) = √ e−t dt. (A.17) π x Properties: erf(−x) =−erf(x) (A.18) erf(x∗) = erf(x)∗ (A.19) 1 ∞ u2 Q(x) = √ exp − du (A.20) 2π x 2 % √ & erfc(x) = 2Q x 2 (A.21) 1 x Q(x) = erfc √ . (A.22) 2 2 The Taylor series expansion of the error function, which provides a good approx- imation of erf(x) for 60 terms in the summation and for x up to 4, is given by ∞ 2 (−1)n x2n+1 erf(x) = √ . (A.23) π n!(2n + 1) n=0 A Mathematical Tables and Algorithms 843 The complementary error function has an alternative representation which is par- ticularly useful in error probability analysis of communication systems. It is known as the desired form of the complementary error function and is given by π/2 2 = 1 − 2x θ. erfc(x) exp 2 d (A.24) π 0 2sin θ A.2.1 Some Tabulated Values of the Complementary Error Function x erfc(x) x erfc(x) x erfc(x) x erfc(x) 0.025 0.971796 1.025 0.147179 2.025 4.19E-03 3.025 1.89E-05 0.05 0.943628 1.05 0.137564 2.05 3.74E-03 3.05 1.61E-05 0.075 0.91553 1.075 0.128441 2.075 3.34E-03 3.075 1.37E-05 0.1 0.887537 1.1 0.119795 2.1 2.98E-03 3.1 1.16E-05 0.125 0.859684 1.125 0.111612 2.125 2.65E-03 3.125 9.90E-06 0.15 0.832004 1.15 0.103876 2.15 2.36E-03 3.15 8.40E-06 0.175 0.804531 1.175 0.096573 2.175 2.10E-03 3.175 7.12E-06 0.2 0.777297 1.2 0.089686 2.2 1.86E-03 3.2 6.03E-06 0.225 0.750335 1.225 0.0832 2.225 1.65E-03 3.225 5.09E-06 0.25 0.723674 1.25 0.0771 2.25 1.46E-03 3.25 4.30E-06 0.275 0.697344 1.275 0.071369 2.275 1.29E-03 3.275 3.63E-06 0.3 0.671373 1.3 0.065992 2.3 1.14E-03 3.3 3.06E-06 0.325 0.645789 1.325 0.060953 2.325 1.01E-03 3.325 2.57E-06 0.35 0.620618 1.35 0.056238 2.35 8.89E-04 3.35 2.16E-06 0.375 0.595883 1.375 0.05183 2.375 7.83E-04 3.375 1.82E-06 0.4 0.571608 1.4 0.047715 2.4 6.89E-04 3.4 1.52E-06 0.425 0.547813 1.425 0.043878 2.425 6.05E-04 3.425 1.27E-06 0.45 0.524518 1.45 0.040305 2.45 5.31E-04 3.45 1.07E-06 0.475 0.501742 1.475 0.036982 2.475 4.65E-04 3.475 8.91E-07 0.5 0.4795 1.5 0.033895 2.5 4.07E-04 3.5 7.43E-07 0.525 0.457807 1.525 0.031031 2.525 3.56E-04 3.525 6.19E-07 0.55 0.436677 1.55 0.028377 2.55 3.11E-04 3.55 5.15E-07 0.575 0.416119 1.575 0.025921 2.575 2.71E-04 3.575 4.29E-07 0.6 0.396144 1.6 0.023652 2.6 2.36E-04 3.6 3.56E-07 0.625 0.376759 1.625 0.021556 2.625 2.05E-04 3.625 2.95E-07 0.65 0.357971 1.65 0.019624 2.65 1.78E-04 3.65 2.44E-07 0.675 0.339783 1.675 0.017846 2.675 1.55E-04 3.675 2.02E-07 0.7 0.322199 1.7 0.01621 2.7 1.34E-04 3.7 1.67E-07 0.725 0.305219 1.725 0.014707 2.725 1.16E-04 3.725 1.38E-07 0.75 0.288844 1.75 0.013328 2.75 1.01E-04 3.75 1.14E-07 0.775 0.273072 1.775 0.012065 2.775 8.69E-05 3.775 9.36E-08 844 A Mathematical Tables and Algorithms x erfc(x) x erfc(x) x erfc(x) x erfc(x) 0.8 0.257899 1.8 0.010909 2.8 7.50E-05 3.8 7.70E-08 0.825 0.243321 1.825 9.85E-03 2.825 6.47E-05 3.825 6.32E-08 0.85 0.229332 1.85 8.89E-03 2.85 5.57E-05 3.85 5.19E-08 0.875 0.215925 1.875 8.01E-03 2.875 4.79E-05 3.875 4.25E-08 0.9 0.203092 1.9 7.21E-03 2.9 4.11E-05 3.9 3.48E-08 0.925 0.190823 1.925 6.48E-03 2.925 3.53E-05 3.925 2.84E-08 0.95 0.179109 1.95 5.82E-03 2.95 3.02E-05 3.95 2.32E-08 0.975 0.167938 1.975 5.22E-03 2.975 2.58E-05 3.975 1.89E-08 1 0.157299 2 4.68E-03 3 2.21E-05 4 1.54E-08 A.3 Derivatives d dw dv du (uvw) = uv + uw + vw (A.25) dx dx dx dx % & d u 1 du dv = v − u (A.26) dx v v2 dx dx d du dv (uv) = vuv−1 + (ln u)uv (A.27) dx dx dx d du dv (uv) = vuv−1 + (ln u)uv (A.28) dx dx dx d 1 du (log u) = (log e) (A.29) dx a a u dx − dn n dnu n dv dn 1u n dnv (uv) = v + +···+ u (A.30) n 0 n 1 n−1 n n dx dx dx dx dx d x f (t)dt = f (x) (A.31) dx c d g(x) g(x) ∂h(x, t) h(x, t)dt = g(x)h(x, g(x)) − f (x)h(x, f (x)) + dt dx f (x) f (x) ∂x (A.32) f (x) f (x) lim = lim . (A.33) x→a g(x) x→a g(x) A.4 Definite Integrals ∞ √ 1√ xe−x dx = π (A.34) 0 2 ∞ 2 1√ e−x dx = π (A.35) 0 2 A Mathematical Tables and Algorithms 845 ∞ x z−1 e−x dx = Γ(z) (A.36) 0 ∞ a e−ax cos bxdx = (A.37) 2 + 2 0 a b ∞ a e−ax sin bxdx = (A.38) a2 + b2 0 ∞ e−ax sin bx b dx = tan−1 (A.39) x a 0 ∞ e−ax − e−bx b dx = ln (A.40) 0 x a ∞ 2 1 π e−ax dx = (A.41) 0 2 a ∞ 2 1 π 2 e−ax cos bxdx = e−b /4a (A.42) 0 2 a ∞ Γ(n + 1) xne−axdx = , a > 0, n > −1 an+1 0 (A.43) n! = , a > 0, n = 0, 1, 2,... an+1 ∞ [1 − erf(βx)] exp −μx2 xdx 0 ' ( (A.44) 1 β = 1 − , Reμ>−Reβ2, Reμ>0 2μ μ + β2 2π x cos θ e dθ = 2π I0(x) (A.45) 0 ∞ π 2 −(ax2+bx+c) b −4ac e dx = e 4a (A.46) −∞ a √ ∞ νπΓ(ν/2) (1 + x2/ν)−(ν+1)/2dx = .