## [Maxima-commits] CVS: maxima/tests rtest14.mac,1.22,1.23

 [Maxima-commits] CVS: maxima/tests rtest14.mac,1.22,1.23 From: Raymond Toy - 2005-02-28 23:16:54 ```Update of /cvsroot/maxima/maxima/tests In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv18990/tests Modified Files: rtest14.mac Log Message: o Add more comments and derivations. o Add a test from Tables of Integral Transforms, formula (42), Section 4.14. Index: rtest14.mac =================================================================== RCS file: /cvsroot/maxima/maxima/tests/rtest14.mac,v retrieving revision 1.22 retrieving revision 1.23 diff -u -d -r1.22 -r1.23 --- rtest14.mac 28 Feb 2005 18:12:31 -0000 1.22 +++ rtest14.mac 28 Feb 2005 23:16:45 -0000 1.23 @@ -395,7 +395,13 @@ /* - * t^(3/2)*bessel_j(1,a*t) -> gamma(3/2+1+1)*r^(-3/2-1)*assoc_legendre_p(3/2,-1,p/r) + * Formula 48, p 187: + * + * t^u*bessel_y(v,a*t) + * -> r^(-u-1)*(gamma(u+v+1)*cot(v*%pi)*assoc_legendre_p(u,-v,p/r) + * -gamma(u-v+1)*csc(v*%pi)*assoc_legendre_p(u,v,p/r)) + * + * But this doesn't work for v = 1. */ specint( t^(3/2)*bessel_y(1,a*t)*%e^(-t),t); 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))/ @@ -526,6 +532,31 @@ * (34) * 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) + * + * A&S 13.1.32 gives + * + * %m[k,u](z) = exp(-z/2)*z^(u+1/2)*M(1/2+u-k,1+2*u,z) + * + * A&S 13.1.27 (Kummer Transformation): + * + * M(a,b,z) = exp(z)*M(b-a,b,-z) + * + * So + * + * %m[k,u](z) = exp(z/2)*z^(u+1/2)*M(1/2+u+k,1+2*u,-z) + * + * But %m[-k,u](-z) = exp(z/2)*(-z)^(u+1/2)*M(1/2+u+k,1+2*u,-z) + * + * Therefore + * + * %m[k,u](z) = (-1)^(u+1/2)*%m[-k,u](-z) + * + * So the Laplace transform can also be written as + * + * gamma(u+v+1/2)/sqrt(a)/gamma(2*v+1)*p^(-u)*exp(-a/p/2) + * *%m[-u,v](-a/p)*(-1)^(v+1/2) + * + * Which is the answer we produce. */ prefer_whittaker:true; true\$ @@ -562,3 +593,19 @@ 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))\$ + +/* + * (42) + * + * t^(lam-1)*bessel_j(2*u,2*sqrt(a)*sqrt(t))*bessel_j(2*v,2*sqrt(a)*sqrt(t)) -> + * gamma(lam+u+v)/gamma(2*u+1)/gamma(2*v+1)*a^(u+v)/p^(lam+u+v) + * *%f[3,3]([u+v+1/2,u+v+1,lam+u+v],[2*u+1,2*v+1,2*u+2*v+1],-4*a/p) + * + */ +(assume(u>0,v>0,lam>0),true); +true\$ +specint(t^(lam-1)*bessel_j(2*u,2*sqrt(a)*sqrt(t))*bessel_j(2*v,2*sqrt(a)*sqrt(t))*exp(-p*t),t); +a^(v+u)*p^(-v-u-lam)*gamma(v+u+lam) + *%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))\$ + ```

 [Maxima-commits] CVS: maxima/tests rtest14.mac,1.22,1.23 From: Raymond Toy - 2005-02-28 23:16:54 ```Update of /cvsroot/maxima/maxima/tests In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv18990/tests Modified Files: rtest14.mac Log Message: o Add more comments and derivations. o Add a test from Tables of Integral Transforms, formula (42), Section 4.14. Index: rtest14.mac =================================================================== RCS file: /cvsroot/maxima/maxima/tests/rtest14.mac,v retrieving revision 1.22 retrieving revision 1.23 diff -u -d -r1.22 -r1.23 --- rtest14.mac 28 Feb 2005 18:12:31 -0000 1.22 +++ rtest14.mac 28 Feb 2005 23:16:45 -0000 1.23 @@ -395,7 +395,13 @@ /* - * t^(3/2)*bessel_j(1,a*t) -> gamma(3/2+1+1)*r^(-3/2-1)*assoc_legendre_p(3/2,-1,p/r) + * Formula 48, p 187: + * + * t^u*bessel_y(v,a*t) + * -> r^(-u-1)*(gamma(u+v+1)*cot(v*%pi)*assoc_legendre_p(u,-v,p/r) + * -gamma(u-v+1)*csc(v*%pi)*assoc_legendre_p(u,v,p/r)) + * + * But this doesn't work for v = 1. */ specint( t^(3/2)*bessel_y(1,a*t)*%e^(-t),t); 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))/ @@ -526,6 +532,31 @@ * (34) * 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) + * + * A&S 13.1.32 gives + * + * %m[k,u](z) = exp(-z/2)*z^(u+1/2)*M(1/2+u-k,1+2*u,z) + * + * A&S 13.1.27 (Kummer Transformation): + * + * M(a,b,z) = exp(z)*M(b-a,b,-z) + * + * So + * + * %m[k,u](z) = exp(z/2)*z^(u+1/2)*M(1/2+u+k,1+2*u,-z) + * + * But %m[-k,u](-z) = exp(z/2)*(-z)^(u+1/2)*M(1/2+u+k,1+2*u,-z) + * + * Therefore + * + * %m[k,u](z) = (-1)^(u+1/2)*%m[-k,u](-z) + * + * So the Laplace transform can also be written as + * + * gamma(u+v+1/2)/sqrt(a)/gamma(2*v+1)*p^(-u)*exp(-a/p/2) + * *%m[-u,v](-a/p)*(-1)^(v+1/2) + * + * Which is the answer we produce. */ prefer_whittaker:true; true\$ @@ -562,3 +593,19 @@ 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))\$ + +/* + * (42) + * + * t^(lam-1)*bessel_j(2*u,2*sqrt(a)*sqrt(t))*bessel_j(2*v,2*sqrt(a)*sqrt(t)) -> + * gamma(lam+u+v)/gamma(2*u+1)/gamma(2*v+1)*a^(u+v)/p^(lam+u+v) + * *%f[3,3]([u+v+1/2,u+v+1,lam+u+v],[2*u+1,2*v+1,2*u+2*v+1],-4*a/p) + * + */ +(assume(u>0,v>0,lam>0),true); +true\$ +specint(t^(lam-1)*bessel_j(2*u,2*sqrt(a)*sqrt(t))*bessel_j(2*v,2*sqrt(a)*sqrt(t))*exp(-p*t),t); +a^(v+u)*p^(-v-u-lam)*gamma(v+u+lam) + *%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))\$ + ```