Sun Apr 5 00:05:13 MET DST 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Special Functions ---------- -- Bernoulli numbers: B_16 => -3617/510 [Gradshteyn and Ryzhik 9.71] bernoulli(16) 3617 (1) - ---- 510 Type: Fraction Integer Time: 0.02 (IN) + 0.07 (EV) + 0.03 (OT) + 0.03 (GC) = 0.15 sec -- 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)] --D(E(phi, k), k) -- Jacobian elliptic functions: d/du dn u => -k^2 sn u cn u -- [Gradshteyn and Ryzhik 8.158(3)] --D(dn(u), u) -- => -2 sqrt(pi) [Gradshteyn and Ryzhik 8.338(3)] Gamma(-1/2) (2) - 3.5449077018110313 Type: DoubleFloat Time: 0.03 (IN) + 0.03 (OT) + 0.02 (GC) = 0.08 sec % + 2*sqrt(%pi) (3) 4.4408920985006262e-16 Type: Expression DoubleFloat Time: 0.68 (IN) + 0.17 (OT) + 0.15 (GC) = 1.0 sec -- 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)] digamma(1/3) (4) - 3.1320337800208065 Type: DoubleFloat Time: 0.02 (IN) = 0.02 sec % + %pi/2*sqrt(1/3) + 3/2*log(3) (5) - 0.57721566490153275 Type: Expression DoubleFloat Time: 0.52 (IN) + 0.10 (EV) + 0.05 (OT) = 0.67 sec -- Bessel function of the first kind of order 2 => 0.04158 + 0.24740 i besselJ(2, 1 + %i) (6) 0.041579886943962127 + 0.2473976415133064%i Type: Complex DoubleFloat Time: 0.12 (IN) + 0.02 (EV) + 0.05 (OT) + 0.03 (GC) = 0.22 sec -- => 12/pi^2 [Gradshteyn and Ryzhik 8.464(6)] besselJ(-5/2, %pi/2) (7) 1.2158542037080535 Type: DoubleFloat Time: 0.07 (IN) + 0.03 (OT) = 0.10 sec % - 12/%pi**2 (8) 2.2204460492503131e-16 Type: DoubleFloat Time: 0.03 (IN) = 0.03 sec -- => sqrt(2/(pi z)) (sin z/z - cos z) [Gradshteyn and Ryzhik 8.464(3)] besselJ(3/2, z) 3 (9) besselJ(-,z) 2 Type: Expression Integer Time: 0.30 (IN) + 0.02 (EV) + 0.02 (OT) = 0.33 sec -- d/dz J_0(z) => - J_1(z) [Gradshteyn and Ryzhik 8.473(4)] D(besselJ(0, z), z) - besselJ(1,z) + besselJ(- 1,z) (10) ------------------------------- 2 Type: Expression Integer Time: 0.05 (IN) + 0.05 (EV) + 0.02 (GC) = 0.12 sec -- 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)] --P(mu, nu, 0) -- P^1_3(x) => -3/2 sqrt(1 - x^2) (5 x^2 - 1) -- [Gradshteyn and Ryzhik 8.813(4)] --P(1, 3, x) -- 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) (11) 0 Type: Polynomial Integer Time: 0.03 (IN) + 30.50 (EV) + 11.38 (GC) = 41.91 sec -- T_n(-1) => (-1)^n [Gradshteyn and Ryzhik 8.944(2)] chebyshevT(n, -1) There are 1 exposed and 0 unexposed library operations named chebyshevT having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op chebyshevT to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named chebyshevT with argument type(s) Variable n Integer -- => arcsin z/z [Gradshteyn and Ryzhik 9.121(26)] --hypergeometric([1/2, 1/2], [3/2], z**2) -- => 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) -- zeta'(0) => - 1/2 log(2 pi) [Gradshteyn and Ryzhik 9.542(4)] --subst(D(zeta(x), x), x = 0) -- Dirac delta distribution => 3 f(4/5) + g'(1) --f:= operator('f); --g:= operator('g); --integrate(f((x + 2)/5)*delta((x - 2)/3) - g(x)*D(delta(x - 1), x), x = 0..3) --)clear properties f g -- Define an antisymmetric function f f:= operator('f); Type: BasicOperator Time: 0.07 (IN) = 0.07 sec -- Test it out => [-f(a, b, c), 0] [f(c, b, a), f(c, b, c)] (13) [f(c,b,a),f(c,b,c)] Type: List Expression Integer Time: 0.17 (IN) + 0.02 (OT) = 0.18 sec )clear properties f -- ---------- Quit ---------- )quit real 68.3 user 34.3 sys 0.4