Tue Feb 23 21:54:05 MST 1999 aquarius% macsyma Starting Macsyma math engine with no window system... /opt/local/macsyma_422/macsyma.422 local This is Macsyma 422.0 for Sparc (Solaris 2.x) computers. Copyright (c) 1982 - 1998 Macsyma Inc. All rights reserved. Portions copyright (c) 1982 Massachusetts Institute of Technology. All rights reserved. Type "DESCRIBE(TRADE_SECRET);" to see important legal notices. Type "HELP();" for more information. /aquarius/data2/opt/local/macsyma_422/system/init.lsp being loaded. /aquarius/home/wester/macsyma-init.lsp being loaded. (c1) (c2) /* ----------[ M a c s y m a ]---------- */ /* ---------- Initialization ---------- */ symbol_display_case: lower_case$ Time= 0 msecs (c3) showtime: all$ Time= 0 msecs (c4) prederror: false$ Time= 0 msecs (c5) /* ---------- Special Functions ---------- */ /* Bernoulli numbers: B_16 => -3617/510 [Gradshteyn and Ryzhik 9.71] */ bern(16); /aquarius/data2/opt/local/macsyma_422/library1/combin.so being loaded. Time= 250 msecs 3617 (d5) - ---- 510 (c6) /* 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)] */ diff(elliptic_e(phi, k^2), k); Time= 10 msecs 2 2 elliptic_e(phi, k ) elliptic_f(phi, k ) (d6) 2 k (------------------- - -------------------) 2 2 2 k 2 k (c7) multthru(%); Time= 0 msecs 2 2 elliptic_e(phi, k ) elliptic_f(phi, k ) (d7) ------------------- - ------------------- k k (c8) /* Jacobian elliptic functions: d/du dn u => -k^2 sn u cn u [Gradshteyn and Ryzhik 8.158(3)] */ diff(jacobi_dn(u, k^2), u); Time= 0 msecs 2 2 2 (d8) - k jacobi_cn(u, k ) jacobi_sn(u, k ) (c9) /* => -2 sqrt(pi) [Gradshteyn and Ryzhik 8.338(3)] */ gamma(-1/2); Time= 0 msecs (d9) - 2 sqrt(%pi) (c10) /* 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[0](1/3); /aquarius/data2/opt/local/macsyma_422/library1/specfn.so being loaded. Time= 70 msecs 3 log(3) %pi (d10) - -------- - --------- - %gamma 2 2 sqrt(3) (c11) /* Bessel function of the first kind of order 2 => 0.04158 + 0.24740 i */ sfloat(bessel_j[2](1 + %i)); /aquarius/data2/opt/local/macsyma_422/library1/bessel.so being loaded. Time= 90 msecs (d11) 0.2473978 %i + 0.04157966 (c12) /* => 12/pi^2 [Gradshteyn and Ryzhik 8.464(6)] */ bessel_j[-5/2](%pi/2); Time= 10 msecs 12 (d12) ---- 2 %pi (c13) /* => sqrt(2/(pi z)) (sin z/z - cos z) [Gradshteyn and Ryzhik 8.464(3)] */ bessel_j[3/2](z); Time= 10 msecs sin(z) sqrt(2) (------ - cos(z)) z (d13) ------------------------- sqrt(%pi) sqrt(z) (c14) /* d/dz J_0(z) => - J_1(z) [Gradshteyn and Ryzhik 8.473(4)] */ diff(bessel_j[0](z), z); Time= 0 msecs (d14) - bessel_j (z) 1 (c15) /* 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)] */ declare(nu, integer, mu, integer)$ Time= 0 msecs (c16) assume(nu > mu)$ Time= 240 msecs (c17) alegendre_p(nu, mu, 0); /aquarius/data2/opt/local/macsyma_422/share/specfun.so being loaded. /aquarius/data2/opt/local/macsyma_422/share/specfun2.so being loaded. Time= 480 msecs mu nu mu nu + mu + 1 2 cos(%pi (-- + --)) gamma(-----------) 2 2 2 (d17) ----------------------------------------- nu mu sqrt(%pi) gamma(-- - -- + 1) 2 2 (c18) forget(nu > mu)$ Time= 0 msecs (c19) remove(nu, integer, mu, integer)$ Time= 0 msecs (c20) /* P^1_3(x) => -3/2 sqrt(1 - x^2) (5 x^2 - 1) [Gradshteyn and Ryzhik 8.813(4)] */ alegendre_p(3, 1, x); /aquarius/data2/opt/local/macsyma_422/library1/binoml.so being loaded. Time= 80 msecs 2 2 sqrt(1 - x ) (15 x - 3) (d20) - ------------------------ 2 (c21) /* nth Chebyshev polynomial of the 1st kind: T_n(x) => 0 [Gradshteyn and Ryzhik 8.941(1)] */ ratsimp(chebyshev_t(1008, x) - 2*x*chebyshev_t(1007, x) + chebyshev_t(1006, x)); Time= 112480 msecs (d21) 0 (c22) /* T_n(-1) => (-1)^n [Gradshteyn and Ryzhik 8.944(2)] */ declare(n, integer)$ Time= 0 msecs (c23) assume(n > 0)$ Time= 0 msecs (c24) chebyshev_t(n, -1); Time= 80 msecs n (d24) (- 1) (c25) forget(n > 0)$ Time= 0 msecs (c26) assume(equal(n, 0))$ Time= 10 msecs (c27) chebyshev_t(n, -1); Time= 30 msecs n (d27) (- 1) (c28) forget(equal(n, 0))$ Time= 0 msecs (c29) assume(n < 0)$ Time= 0 msecs (c30) chebyshev_t(n, -1); Time= 80 msecs (d30) chebyshev_t(n, - 1) (c31) forget(n < 0)$ Time= 10 msecs (c32) remove(n, integer)$ Time= 0 msecs (c33) /* => arcsin z/z [Gradshteyn and Ryzhik 9.121(26)] */ hgfred([1/2, 1/2], [3/2], z^2); /aquarius/data2/opt/local/macsyma_422/library1/hyp.so being loaded. Time= 290 msecs asin(z) (d33) ------- z (c34) /* => sin(n z)/(n sin z cos z) [Gradshteyn and Ryzhik 9.121(17)] */ q: hgfred([(n + 2)/2, -(n - 2)/2], [3/2], sin(z)^2); Time= 150 msecs sin((n + 1) asin(sin(z))) cos((n + 1) asin(sin(z))) (d34) ------------------------- - ------------------------- n sin(z) 2 n sqrt(1 - sin (z)) (c35) assume(cos(z) > 0)$ Time= 10 msecs (c36) ev(trigreduce(trigsimp(q)), triginverses:all); /aquarius/data2/opt/local/macsyma_422/share/trigsimp.so being loaded. /aquarius/data2/opt/local/macsyma_422/library1/trgred.so being loaded. Time= 720 msecs 2 csc(2 z) sin(n z) (d36) ------------------- n (c37) forget(cos(z) > 0)$ Time= 0 msecs (c38) remvalue(q)$ Time= 0 msecs (c39) /* zeta'(0) => - 1/2 log(2 pi) [Gradshteyn and Ryzhik 9.542(4)] */ subst(x = 0, diff(zeta(x), x)); Time= 10 msecs log(%pi) + log(2) (d39) - ----------------- 2 (c40) logcontract(%); Time= 0 msecs log(2 %pi) (d40) - ---------- 2 (c41) /* Dirac delta distribution => 3 f(4/5) + g'(1) */ integrate(f((x + 2)/5)*delta((x - 2)/3) - g(x)*diff(delta(x - 1), x), x, 0, 3); Time= 50 msecs 3 / [ x - 2 x + 2 d (d41) I (delta(-----) f(-----) - g(x) (-- (delta(x - 1)))) dx ] 3 5 dx / 0 (c42) delint(f((x + 2)/5)*delta((x - 2)/3) - g(x)*diff(delta(x - 1), x), x, 0, 3); /aquarius/data2/opt/local/macsyma_422/share/delint.so being loaded. /aquarius/data2/opt/local/macsyma_422/share/algfuncs.so being loaded. Time= 280 msecs | d | 4 (d42) -- (g(x))| - delta(2) g(3) + 3 f(-) + delta(- 1) g(0) dx | 5 |x = 1 (c43) deltasimp(%, x); Time= 10 msecs | d | 4 (d43) -- (g(x))| + 3 f(-) dx | 5 |x = 1 (c44) /* Define an antisymmetric function f */ declare(f, antisymmetric)$ Time= 0 msecs (c45) /* Test it out => [-f(a, b, c), 0] */ [f(c, b, a), f(c, b, c)]; Time= 0 msecs (d45) [- f(a, b, c), 0] (c46) remove(f, antisymmetric)$ Time= 0 msecs (c47) /* ---------- Quit ---------- */ quit(); Bye. real 120.26 user 116.05 sys 1.92