From: Raymond T. <rt...@us...> - 2005-03-02 14:26:33
|
Update of /cvsroot/maxima/maxima/tests In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv30247/tests Modified Files: rtest14.mac Log Message: o Document specint( t*%h[2/3,1](t^(1/2))*%e^(-p*t),t); o Added test for bessel_y(v,a*t), since we can handle this now. o Added tests (which currently fail): - bessel_y(0,a*t) - t*bessel_y(0,a*t) - t*bessel_y(1,a*t) Index: rtest14.mac =================================================================== RCS file: /cvsroot/maxima/maxima/tests/rtest14.mac,v retrieving revision 1.23 retrieving revision 1.24 diff -u -d -r1.23 -r1.24 --- rtest14.mac 28 Feb 2005 23:16:45 -0000 1.23 +++ rtest14.mac 2 Mar 2005 14:26:23 -0000 1.24 @@ -321,6 +321,27 @@ factor(ratsimp(specint(t^(1/2)*bessel_i(1,t)*%e^(-p*t),t))); 3*sqrt(%pi)*assoc_legendre_p(1/2,-1,p/sqrt(p^2-1))/(4*(p^2-1)^(3/4))$ +/* + * %h[2/3,1](sqrt(t)) = bessel_j(2/3,sqrt(t))+%i*bessel_y(2/3,sqrt(t)) + * + * Formula (34) below gives: + * + * t^(u-1/2)*bessel_j(2*v,2*sqrt(a)*sqrt(t)) -> + * gamma(u+v+1/2)/sqrt(a)/gamma(2*v+1)*p^(-u)*exp(-a/p/2)*%m[u,v](a/p) + * + * Formula (50) gives + * + * t^(u-1/2)*bessel_y(2*v,2*sqrt(a)*sqrt(t)) -> + * 1/sqrt(a)*p^(-u)*exp(-a/p/2)* + * (tan((u-v)*%pi)*gamma(u+v-1/2)/gamma(2*v+1) * %m[u,v](a/p) + * - sec((u-v)*%pi)*%w[u,v](a/p)) + * + * But A&S 13.1.34 says + * + * %w[k,u](z) = gamma(-2*u)/gamma(1/2-u-k)*%m[k,u](z) + * + gamma(2*u)/gamma(1/2+u-k)*%m[k,-u](z) + * + */ specint( t*%h[2/3,1](t^(1/2))*%e^(-p*t),t); -4*%i*gamma(1/3)*%m[-3/2,1/3](-1/(4*p))*%e^-(1/(8*p))/(3*(-1)^(5/6)*sqrt(3) *gamma(2/3)*p^(3/2))+4*gamma(1/3)*%m[-3/2,1/3](-1/(4*p))*%e^-(1/(8*p))/(3*( @@ -407,6 +428,7 @@ 15*%i*(1/(a^2+1)-1)^(3/4)*(1/(sqrt(a^2+1)+1)^(3/2)+1/(1-sqrt(a^2+1))^(3/2))/ (16*sqrt(a^2+1)) $ + /* * Duplicate */ @@ -586,13 +608,17 @@ * a^v*cot(v*%pi)/r*R^(-v)-a^(-v)*csc(v*%pi)/r*R^v * For |Re v| < 1. * - * But maxima currently doesn't understand this for general v, so use - * some specific values. */ expand(factor(radcan(specint(exp(-p*t)*bessel_y(1/6,a*t),t)))); sqrt(3)*a^(1/6)/(sqrt(p^2+a^2)*(sqrt(p^2+a^2)+p)^(1/6)) -2*(sqrt(p^2+a^2)+p)^(1/6)/(a^(1/6)*sqrt(p^2+a^2))$ +(assume(v1 > 0, v1 < 1), true); +true$ +expand(factor(radcan(specint(exp(-p*t)*bessel_y(v1,a*t),t)))); +a^v1*cot(%pi*v1)/(sqrt(p^2+a^2)*(sqrt(p^2+a^2)+p)^v1) + -(sqrt(p^2+a^2)+p)^v1/(a^v1*sqrt(p^2+a^2)*sin(%pi*v1)) $ + /* * (42) @@ -609,3 +635,30 @@ *%f[3,3]([v+u+1/2,v+u+1,v+u+lam],[2*u+1,2*v+1,2*v+2*u+1],-4*a/p) /(gamma(2*u+1)*gamma(2*v+1))$ + +/* + * (44) + * + * bessel_y(0,a*t) -> -2/%pi/sqrt(p^2+a^2)*asinh(p/a) + */ +specint(bessel_y(0,a*t)*exp(-p*t),t); +-2/%pi/sqrt(p^2+a^2)*asinh(p/a) $ + +/* + * (46) + * + * t*bessel_y(0,a*t) + * -> 2/%pi/(p^2+a^2)*(1-p/sqrt(p^2+a^2)*log((p+sqrt(p^2+a^2))/a) + */ +specint(t*bessel_y(0,a*t)*exp(-p*t),t); +2/%pi/(p^2+a^2)*(1-p/sqrt(p^2+a^2)*log((p+sqrt(p^2+a^2))/a)) $ + +/* + * (47) + * + * t*bessel_y(1,a*t) + * -> -2/%pi/(p^2+a^2)*(p/a-a/sqrt(p^2+a^2)*log((p+sqrt(p^2+a^2))/a) + */ +specint(t*bessel_y(1,a*t)*exp(-p*t),t); +-2/%pi/(p^2+a^2)*(p/a-a/sqrt(p^2+a^2)*log((p+sqrt(p^2+a^2))/a)) $ + |