Thu Feb 25 22:58:40 MST 1999 aquarius% maple |\^/| Maple V Release 5.1 (WMI Campus Wide License) ._|\| |/|_. Copyright (c) 1981-1998 by Waterloo Maple Inc. All rights \ MAPLE / reserved. Maple and Maple V are registered trademarks of <____ ____> Waterloo Maple Inc. | Type ? for help. # ----------[ M a p l e ]---------- #interface(echo = 3); # ---------- Initialization ---------- > readlib(showtime): > on; # ---------- Special Functions ---------- # Bernoulli numbers: B_16 => -3617/510 [Gradshteyn and Ryzhik 9.71] O1 := bernoulli(16); -3617 ----- 510 time = 0.04, bytes = 39554 # d/dk E(phi, k) => [E(phi, k) - F(phi, k)]/k where F(phi, k) and E(phi, k) # are elliptic integrals of the 1st and 2nd kind, respectively # [Gradshteyn and Ryzhik 8.123(3)] O2 := diff(EllipticE(sin(phi), k), k); EllipticF(sin(phi), k) EllipticE(sin(phi), k) - ---------------------- + ---------------------- k k time = 0.16, bytes = 313846 # Jacobian elliptic functions: d/du dn u => -k^2 sn u cn u # [Gradshteyn and Ryzhik 8.158(3)] O3 := diff(JacobiDN(u, k), u); 2 -k JacobiCN(u, k) JacobiSN(u, k) time = 0.04, bytes = 27162 # => -2 sqrt(pi) [Gradshteyn and Ryzhik 8.338(3)] O4 := GAMMA(-1/2); 1/2 -2 Pi time = 0.02, bytes = 28966 # psi(1/3) => - Euler's_constant - pi/2 sqrt(1/3) - 3/2 log 3 where psi(x) # is the psi function [= d/dx log Gamma(x)] [Gradshteyn and Ryzhik 8.366(6)] O5 := Psi(1/3); 1/2 -gamma - 1/6 Pi 3 - 3/2 ln(3) time = 0.02, bytes = 18718 # Bessel function of the first kind of order 2 => 0.04158 + 0.24740 i O6 := evalf(BesselJ(2, 1 + I)); .04157988694 + .2473976415 I time = 0.09, bytes = 95466 # => 12/pi^2 [Gradshteyn and Ryzhik 8.464(6)] O7 := BesselJ(-5/2, Pi/2); 12 --- 2 Pi time = 0.04, bytes = 57890 # => sqrt(2/(pi z)) (sin z/z - cos z) [Gradshteyn and Ryzhik 8.464(3)] O8 := BesselJ(3/2, z); 1/2 2 (cos(z) z - sin(z)) - ------------------------ 1/2 3/2 Pi z time = 0.03, bytes = 40850 # d/dz J_0(z) => - J_1(z) [Gradshteyn and Ryzhik 8.473(4)] O9 := diff(BesselJ(0, z), z); -BesselJ(1, z) time = 0.03, bytes = 27354 # Associated Legendre (spherical) function of the 1st kind: P^mu_nu(0) # => 2^mu sqrt(pi) / [Gamma([nu - mu]/2 + 1) Gamma([- nu - mu + 1]/2)] # [Gradshteyn and Ryzhik 8.756(1)] O10 := LegendreP(nu, mu, 0); 1/2 (1/2 mu) mu Pi (-1) 2 ------------------------------------------------------- GAMMA(1/2 nu - 1/2 mu + 1) GAMMA(1/2 - 1/2 mu - 1/2 nu) time = 0.07, bytes = 83878 # P^1_3(x) => -3/2 sqrt(1 - x^2) (5 x^2 - 1) # [Gradshteyn and Ryzhik 8.813(4)] O11 := LegendreP(3, 1, x); LegendreP(3, 1, x) time = 0.01, bytes = 15182 # nth Chebyshev polynomial of the 1st kind: T_n(x) => 0 # [Gradshteyn and Ryzhik 8.941(1)] O12 := simplify(orthopoly[T](1008, x) - 2*x*orthopoly[T](1007, x) O12 := + orthopoly[T](1006, x)); bytes used=1017200, alloc=851812, time=0.77 bytes used=2017476, alloc=1441528, time=1.20 bytes used=3035512, alloc=1638100, time=1.72 bytes used=4043496, alloc=1769148, time=2.45 bytes used=5061900, alloc=1834672, time=3.19 bytes used=6071436, alloc=1834672, time=3.90 bytes used=7093496, alloc=1900196, time=4.66 bytes used=8095860, alloc=1900196, time=5.29 bytes used=9124244, alloc=1900196, time=5.85 bytes used=10128712, alloc=1900196, time=6.49 bytes used=11142344, alloc=2031244, time=7.51 bytes used=12259656, alloc=2227816, time=8.72 bytes used=13262776, alloc=2293340, time=9.82 bytes used=14295708, alloc=2293340, time=10.93 bytes used=15310132, alloc=2293340, time=12.06 bytes used=16326120, alloc=2293340, time=13.19 bytes used=17346744, alloc=2293340, time=14.36 bytes used=18355960, alloc=2293340, time=15.46 bytes used=19370220, alloc=2293340, time=16.62 bytes used=20385388, alloc=2293340, time=17.71 bytes used=21415488, alloc=2293340, time=18.82 bytes used=22471092, alloc=2358864, time=19.95 bytes used=23509472, alloc=2358864, time=20.83 bytes used=24519368, alloc=2358864, time=21.85 bytes used=25553108, alloc=2424388, time=22.95 bytes used=26651080, alloc=2489912, time=24.26 bytes used=27684356, alloc=2489912, time=25.32 bytes used=28690188, alloc=2489912, time=26.47 bytes used=29842684, alloc=2555436, time=27.86 bytes used=30877700, alloc=2620960, time=28.97 bytes used=31882412, alloc=2620960, time=30.15 bytes used=32895484, alloc=2620960, time=31.33 bytes used=33937020, alloc=2620960, time=32.48 bytes used=34981592, alloc=2620960, time=33.69 bytes used=35986924, alloc=2620960, time=34.67 bytes used=37003432, alloc=2620960, time=35.74 bytes used=38004968, alloc=2620960, time=36.72 bytes used=39027532, alloc=2620960, time=37.69 bytes used=40039348, alloc=2620960, time=38.63 bytes used=41039592, alloc=2620960, time=39.77 bytes used=42065436, alloc=2620960, time=40.81 bytes used=43088240, alloc=2620960, time=41.92 bytes used=44092408, alloc=2620960, time=42.89 bytes used=45103116, alloc=2620960, time=43.69 bytes used=46188624, alloc=2620960, time=44.20 bytes used=47194656, alloc=2620960, time=44.77 bytes used=48196980, alloc=2620960, time=45.44 bytes used=49204852, alloc=2620960, time=46.34 bytes used=50212792, alloc=2620960, time=47.39 bytes used=51230740, alloc=2620960, time=48.18 bytes used=52235092, alloc=2620960, time=49.10 bytes used=53238820, alloc=2620960, time=49.94 bytes used=54246892, alloc=2620960, time=50.56 bytes used=55260736, alloc=2620960, time=51.17 bytes used=56265540, alloc=2620960, time=51.93 bytes used=57269104, alloc=2620960, time=52.97 bytes used=58284024, alloc=2620960, time=54.09 bytes used=59293200, alloc=2620960, time=54.93 bytes used=60357504, alloc=2620960, time=55.78 bytes used=61373204, alloc=2620960, time=56.58 bytes used=62377744, alloc=2620960, time=57.28 bytes used=63401004, alloc=2620960, time=57.90 bytes used=64435144, alloc=2620960, time=59.29 bytes used=65471404, alloc=2620960, time=60.57 bytes used=66475232, alloc=2752008, time=61.97 bytes used=67553964, alloc=2817532, time=63.51 bytes used=68666984, alloc=2883056, time=65.18 bytes used=69689564, alloc=2948580, time=66.69 bytes used=70692848, alloc=2948580, time=68.17 bytes used=71700360, alloc=2948580, time=69.69 bytes used=72711428, alloc=3014104, time=71.27 bytes used=73711676, alloc=3014104, time=72.80 bytes used=74830340, alloc=3014104, time=74.54 bytes used=75954456, alloc=3145152, time=76.23 bytes used=76958632, alloc=3145152, time=77.27 bytes used=77996684, alloc=3145152, time=78.53 bytes used=79019448, alloc=3145152, time=79.67 bytes used=80058760, alloc=3145152, time=81.15 bytes used=81083184, alloc=3145152, time=82.27 bytes used=82146408, alloc=3145152, time=83.48 bytes used=83150048, alloc=3145152, time=84.55 bytes used=84170532, alloc=3145152, time=85.90 bytes used=85184572, alloc=3145152, time=87.23 bytes used=86187396, alloc=3145152, time=88.54 bytes used=87190428, alloc=3145152, time=89.71 bytes used=88210872, alloc=3145152, time=90.75 bytes used=89212468, alloc=3145152, time=92.05 bytes used=90229136, alloc=3145152, time=93.32 bytes used=91258432, alloc=3145152, time=94.78 bytes used=92281480, alloc=3145152, time=96.09 bytes used=93309368, alloc=3145152, time=97.37 bytes used=94311172, alloc=3145152, time=98.30 bytes used=95323452, alloc=3145152, time=99.82 bytes used=96359812, alloc=3145152, time=101.11 bytes used=97367516, alloc=3145152, time=102.46 bytes used=98387376, alloc=3145152, time=103.73 bytes used=99459944, alloc=3145152, time=105.29 bytes used=100478368, alloc=3145152, time=106.48 bytes used=101480444, alloc=3145152, time=107.62 bytes used=102482268, alloc=3145152, time=108.60 bytes used=103487344, alloc=3145152, time=109.84 bytes used=104496408, alloc=3145152, time=110.94 bytes used=105497048, alloc=3145152, time=111.86 bytes used=106501492, alloc=3145152, time=112.44 bytes used=107503076, alloc=3145152, time=113.00 bytes used=108506472, alloc=3145152, time=113.91 bytes used=109519688, alloc=3145152, time=115.00 bytes used=110566588, alloc=3145152, time=116.29 bytes used=111632328, alloc=3145152, time=117.28 bytes used=112695476, alloc=3145152, time=118.22 bytes used=113710260, alloc=3145152, time=119.06 bytes used=114714584, alloc=3145152, time=119.80 bytes used=115718164, alloc=3145152, time=120.54 bytes used=116736092, alloc=3145152, time=121.78 bytes used=117747300, alloc=3145152, time=123.14 bytes used=118750224, alloc=3145152, time=124.60 bytes used=119782328, alloc=3145152, time=126.13 bytes used=120801416, alloc=3145152, time=127.69 bytes used=121814708, alloc=3145152, time=129.25 bytes used=122818188, alloc=3145152, time=130.81 bytes used=123897468, alloc=3145152, time=132.60 bytes used=124905052, alloc=3145152, time=134.24 bytes used=125925828, alloc=3145152, time=135.78 bytes used=126965584, alloc=3145152, time=137.48 bytes used=128006720, alloc=3145152, time=139.08 bytes used=129042304, alloc=3145152, time=140.29 bytes used=130080776, alloc=3145152, time=141.54 bytes used=131095176, alloc=3145152, time=142.70 bytes used=132185928, alloc=3145152, time=144.29 bytes used=133194096, alloc=3145152, time=145.41 bytes used=134246372, alloc=3145152, time=146.66 bytes used=135283836, alloc=3145152, time=147.80 bytes used=136284560, alloc=3145152, time=149.03 bytes used=137305148, alloc=3145152, time=150.42 bytes used=138317620, alloc=3145152, time=151.71 bytes used=139327392, alloc=3145152, time=152.88 bytes used=140346208, alloc=3145152, time=153.96 bytes used=141364912, alloc=3145152, time=155.35 bytes used=142391920, alloc=3145152, time=156.68 bytes used=143403980, alloc=3210676, time=158.19 bytes used=144404204, alloc=3210676, time=159.41 bytes used=145444488, alloc=3210676, time=160.65 bytes used=146462184, alloc=3210676, time=161.84 bytes used=147463316, alloc=3210676, time=163.04 bytes used=148482048, alloc=3210676, time=164.34 bytes used=149506240, alloc=3210676, time=165.59 bytes used=150523708, alloc=3210676, time=166.72 bytes used=151533344, alloc=3210676, time=167.80 bytes used=152563124, alloc=3210676, time=169.03 bytes used=153587744, alloc=3210676, time=170.06 bytes used=154602504, alloc=3210676, time=170.98 bytes used=155606688, alloc=3210676, time=172.15 bytes used=156665512, alloc=3210676, time=172.78 0 time = 172.43, bytes = 156201598 # T_n(-1) => (-1)^n [Gradshteyn and Ryzhik 8.944(2)] O13 := orthopoly[T](n, -1); orthopoly[T](n, -1) time = 0.02, bytes = 7054 # => arcsin z/z [Gradshteyn and Ryzhik 9.121(26)] O14 := hypergeom([1/2, 1/2], [3/2], z^2); 2 hypergeom([1/2, 1/2], [3/2], z ) time = 0.02, bytes = 17326 O15 := convert(%, StandardFunctions); arcsin(z) --------- z time = 0.13, bytes = 179274 # => sin(n z)/(n sin z cos z) [Gradshteyn and Ryzhik 9.121(17)] O16 := hypergeom([(n + 2)/2, -(n - 2)/2], [3/2], sin(z)^2); 2 hypergeom([1/2 n + 1, - 1/2 n + 1], [3/2], sin(z) ) time = 0.01, bytes = 10086 O17 := simplify(convert(%, StandardFunctions), symbolic); bytes used=157668800, alloc=3210676, time=173.69 sin(n z) --------------- n sin(z) cos(z) time = 0.80, bytes = 712114 # zeta'(0) => - 1/2 log(2 pi) [Gradshteyn and Ryzhik 9.542(4)] O18 := D(Zeta)(0); - 1/2 ln(2 Pi) time = 0.11, bytes = 109550 # Dirac delta distribution => 3 f(4/5) + g'(1) O19 := int(f((x + 2)/5)*Dirac((x - 2)/3) - g(x)*D(Dirac)(x - 1), x = 0..3); / d \ 3 f(4/5) + | lim -- g(x)| \x -> 1 dx / time = 0.15, bytes = 166522 # Define an antisymmetric function f O20 := f:= table(antisymmetric): time = 0.00, bytes = 8014 # Test it out => [-f(a, b, c), 0] O21 := [f[c, b, a], f[c, b, c]]; [-f[c, a, b], 0] time = 0.01, bytes = 6982 O22 := f:= 'f': time = 0.01, bytes = 3610 # ---------- Quit ---------- O23 := quit bytes used=158451468, alloc=3210676, time=174.38 real 175.92 user 174.43 sys 1.11