Sun Nov 1 14:50:44 MST 1998 aquarius% reduce REDUCE 3.6, 15-Jul-95, patched to 15 Apr 96 ... 1: % ---------[ R e d u c e ]---------- % ---------- Initialization ---------- on time; Time: 0 ms % ---------- Special Functions ---------- load_package(specfn)$ Time: 1490 ms plus GC time: 40 ms % Bernoulli numbers: B_16 => -3617/510 [Gradshteyn and Ryzhik 9.71] Bernoulli(16); - 3617 --------- 510 Time: 20 ms % 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)] df(EllipticE(phi, k^2), k); 3 2 2 2 (2*k *(elliptice(phi,k )*jacobicn(phi,k ) 2 2 2 - jacobicn(phi,k )*jacobidn(phi,k )*jacobisn(phi,k ) 2 2 4 2 2 4 - jacobisn(phi,k ) *k *phi + jacobisn(phi,k ) *phi))/(k - 1) Time: 20 ms % Jacobian elliptic functions: d/du dn u => -k^2 sn u cn u % [Gradshteyn and Ryzhik 8.158(3)] df(Jacobidn(u), u); df(jacobidn(u),u) Time: 0 ms % Bessel function of the first kind of order 2 => 0.04158 + 0.24740 i % => -2 sqrt(pi) [Gradshteyn and Ryzhik 8.338(3)] Gamma(-1/2); - 2*pi ---------- sqrt(pi) Time: 10 ms % 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)] Psi(1/3); 1 psi(---) 3 Time: 0 ms % Bessel function of the first kind of order 2 => 0.04158 + 0.24740 i on complex, rounded; *** Domain mode complex changed to complex-rounded Time: 10 ms BesselJ(2, 1 + i); 0.041579886944 + 0.247397641513*i Time: 330 ms off complex, rounded; *** Domain mode complex-rounded changed to rounded Time: 0 ms % => 12/pi^2 [Gradshteyn and Ryzhik 8.464(6)] BesselJ(-5/2, pi/2); - 5 pi besselj(------,----) 2 2 Time: 10 ms % => sqrt(2/(pi z)) (sin z/z - cos z) [Gradshteyn and Ryzhik 8.464(3)] BesselJ(3/2, z); 3 besselj(---,z) 2 Time: 10 ms % d/dz J_0(z) => - J_1(z) [Gradshteyn and Ryzhik 8.473(4)] df(BesselJ(0, z), z); - besselj(1,z) Time: 10 ms % 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)] LegendreP(nu, mu, 0); mu mu*pi + nu*pi mu + nu + 1 2 *cos(---------------)*gamma(-------------) 2 2 ----------------------------------------------- - mu + nu + 2 sqrt(pi)*gamma(----------------) 2 Time: 40 ms % P^1_3(x) => -3/2 sqrt(1 - x^2) (5 x^2 - 1) % [Gradshteyn and Ryzhik 8.813(4)] LegendreP(3, 1, x); 2 2 3*sqrt( - x + 1)*( - 5*x + 1) --------------------------------- 2 Time: 30 ms % nth Chebyshev polynomial of the 1st kind: T_n(x) => 0 % [Gradshteyn and Ryzhik 8.941(1)] ChebyshevT(1008, x) - 2*x*ChebyshevT(1007, x) + ChebyshevT(1006, x); 0 Time: 308680 ms plus GC time: 55490 ms % T_n(-1) => (-1)^n [Gradshteyn and Ryzhik 8.944(2)] ChebyshevT(n, -1); n ( - 1) Time: 10 ms % => arcsin z/z [Gradshteyn and Ryzhik 9.121(26)] load_package(specfn2)$ Time: 130 ms plus GC time: 70 ms hypergeometric({1/2, 1/2}, {3/2}, z^2); asin(abs(z)) -------------- abs(z) Time: 10 ms % => sin(n z)/(n sin z cos z) [Gradshteyn and Ryzhik 9.121(17)] hypergeometric({(n + 2)/2, -(n - 2)/2}, {3/2}, sin(z)^2); n + 2 - n + 2 3 2 hypergeometric({-------,----------},{---},sin(z) ) 2 2 2 Time: 10 ms % zeta'(0) => - 1/2 log(2 pi) [Gradshteyn and Ryzhik 9.542(4)] sub(x = 0, df(Zeta(x), x)); sub(x=0,df(zeta(x),x)) Time: 10 ms % Dirac delta distribution => 3 f(4/5) + g'(1) %operator f, g; %int(f((x + 2)/5)*delta((x - 2)/3) - g(x)*df(delta(x - 1), x), x, 0, 3); %clear f, g; % Define an antisymmetric function f operator f; Time: 0 ms antisymmetric f; Time: 0 ms % Test it out => [-f(a, b, c), 0] {f(c, b, a), f(c, b, c)}; { - f(a,b,c),0} Time: 0 ms clear f; Time: 0 ms % *** Numerical evaluation of elliptic integrals causes conflicts with other % functions, so numerically check the result of d/dk E(phi, k) now. *** df(EllipticE(phi, k^2), k); 3 2 2 2 (2*k *(elliptice(phi,k )*jacobicn(phi,k ) 2 2 2 - jacobicn(phi,k )*jacobidn(phi,k )*jacobisn(phi,k ) 2 2 4 2 2 4 - jacobisn(phi,k ) *k *phi + jacobisn(phi,k ) *phi))/(k - 1) Time: 20 ms on rounded; Time: 0 ms sub({phi = Pi/2, k = 1/4}, ws); - 0.0483389030682 Time: 870 ms plus GC time: 70 ms sub({phi = Pi/2, k = 1/4}, (EllipticE(phi, k^2) - EllipticF(phi, k^2))/k); - 0.201139864105 Time: 400 ms plus GC time: 70 ms off rounded; Time: 0 ms % ---------- Quit ---------- quit; Quitting real 381.36 user 368.01 sys 1.30