[Maxima-commits] CVS: maxima/tests rtest14.mac,1.12,1.13

[Raymond T. <rt...@us...>]

Update of /cvsroot/maxima/maxima/tests
In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv9060/tests

Modified Files:
	rtest14.mac 
Log Message:
Add some new tests from the Tables of Integral Transforms.


Index: rtest14.mac
===================================================================
RCS file: /cvsroot/maxima/maxima/tests/rtest14.mac,v
retrieving revision 1.12
retrieving revision 1.13
diff -u -d -r1.12 -r1.13
--- rtest14.mac	1 Dec 2004 21:18:40 -0000	1.12
+++ rtest14.mac	11 Jan 2005 19:25:30 -0000	1.13
@@ -125,7 +125,7 @@
 
 specint(t^2*bessel_j(1,a*t)*%e^(-p*t),t);
 3*a/((a^2/p^2+1)^(5/2)*p^4) ;
-specint(t*hstruve[1](t)*%e^(-p*t),t);
+radcan(specint(t*hstruve[1](t)*%e^(-p*t),t));
 16*(2*p*sqrt(p^2+1)+2*p^2)/(9*%pi^(3/2)*p^4*(2*p*sqrt(p^2+1)+2*p^2+1)) $
 radcan(specint(t^(3/2)*hstruve[1](t^(1/2))*%e^(-p*t),t));
 %e^-(1/(4*p))*((140*p^(3/2)+10*sqrt(p))*%e^(1/(4*p))+140*sqrt(%pi)*%i*erf(%i/ 
@@ -189,6 +189,141 @@
 15*(1/sqrt(1/(p-a)+1)-1/((p-a)*(1/(p-a)+1)^(3/2))+3/(4*(p-a)^2*(1/(p-a)+1)^ 
  (5/2)))/(4*(p-a)^(7/2)) ;
 
+/*
+ * Laplace transforms from Tables of Integral Transforms
+ */
+
+/*
+ * p 182, (1)
+ *
+ * bessel_j(v,a*t) -> r^(-1)*(a/R)^v
+ *
+ * where r = sqrt(p^2+a^2) and R = p + r
+ */
+(assume(v>0),true);
+true$
 
+radcan(specint(bessel_j(v,a*t)*exp(-p*t),t));
+a^v/(sqrt(p^2+a^2)*(sqrt(p^2+a^2)+p)^v)$
 
+/*
+ * (5)
+ * bessel_j(v,a*t)/t -> v^(-1)*(a/R)^v
+ *
+ * (Maxima doesn't recognize that gamma(v)/gamma(v+1) is 1/v.)
+ */
+radcan(specint(bessel_j(v,a*t)/t*exp(-p*t),t));
+a^v*gamma(v)/((sqrt(p^2+a^2)+p)^v*gamma(v+1))$
+
+/*
+ * (7)
+ * t^v*bessel_j(v,a*t) -> 2^v/sqrt(%pi)*gamma(v+1/2)*a^v*r^(-2*v-1)
+ *
+ * Maxima doesn't recognize the relationship between gamma(2*v+1) and
+ * gamma(v+1).
+ */
+radcan(specint(t^v*bessel_j(v,a*t)*exp(-p*t),t));
+a^v*gamma(2*v+1)/((p^2+a^2)^((2*v+1)/2)*2^v*gamma(v+1))$
+
+/*
+ * (9)
+ * t^u*bessel_j(v,a*t) -> gamma(u+v+1)*r^(-u-1)*P(u,-v,p/r)
+ */
+(assume(v+u+1>0),true);
+true$
+(assume(a>0),true);
+true$
+
+radcan(specint(t^u*bessel_j(v,a*t)*exp(-p*t),t));
+assoc_legendre_p(-u-1,-v,p/sqrt(p^2+a^2))*gamma(v+u+1)
+	/((p^2+a^2)^((u+1)/2)*(-1)^(v/2))$
+
+/*
+ * (25)
+ * bessel_j(0,2*sqrt(a)*sqrt(t)) -> exp(-a/p)/p
+ */
+specint(bessel_j(0,2*sqrt(a)*sqrt(t))*exp(-p*t),t);
+%e^-(a/p)/p$
+
+/*
+ * (27)
+ * sqrt(t)*bessel_j(1,2*sqrt(a)*sqrt(t)) -> sqrt(a)*p^(-2)*exp(-a/p)
+ */
+specint(sqrt(t)*bessel_j(1,2*sqrt(a)*sqrt(t))*exp(-p*t),t);
+sqrt(a)*%e^-(a/p)/p^2$
+
+/*
+ * (29)
+ * t^(-1/2)*bessel_j(1,2*sqrt(a)*sqrt(t)) ->
+ *    sqrt(%pi)/sqrt(p)*exp(-a/2/p)*bessel_i(v/2,a/2/p)
+ */
+specint(t^(-1/2)*bessel_j(1,2*sqrt(a)*sqrt(t))*exp(-p*t),t);
+sqrt(%pi)*bessel_i(1/2,a/(2*p))*%e^-(a/(2*p))/sqrt(p)$
+
+/*
+ * (30)
+ * t^(v/2)*bessel_j(v,2*sqrt(a)*sqrt(t)) ->
+ *    a^(v/2)/p^(v+1)*exp(-a/p)
+ */
+specint(t^(v/2)*bessel_j(v,2*sqrt(a)*sqrt(t))*exp(-p*t),t);
+a^(v/2)*p^(-v-1)*%e^-(a/p)$
+
+/*
+ * (31)
+ * t^(-v/2)*bessel_j(v,2*sqrt(a)*sqrt(t)) ->
+ *    exp(%i*v*%pi)*p^(v-1)/a^(v/2)/gamma(v)*exp(-a/p)*
+ *     %gammagreek(v,a/p*exp(-%i*%pi)
+ *
+ * %gammagreek is the incomplete gamma function.
+ */
+specint(t^(-v/2)*bessel_j(v,2*sqrt(a)*sqrt(t))*exp(-p*t),t);
+p^(v-1)*%e^-(a/p)*v*%gammagreek(v,-a/p)/(a^(v/2)*(-1)^v*gamma(v+1))$
+
+/*
+ * (32)
+ * t^(v/2-1)*bessel_j(v,2*sqrt(a)*sqrt(t)) ->
+ *    a^(-v/2)*%gammagreek(v,a/p)
+ */
+specint(t^(v/2-1)*bessel_j(v,2*sqrt(a)*sqrt(t))*exp(-p*t),t);
+v*gamma(v)*%gammagreek(v,a/p)/(a^(v/2)*gamma(v+1))$
+
+/*
+ * (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)
+ */
+prefer_whittaker:true;
+true$
+(assume(2*v+2*u+1>0),true);
+true$
+
+specint(t^(u-1/2)*bessel_j(2*v,2*sqrt(a)*sqrt(t))*exp(-p*t),t);
+%m[-u,v](-a/p)*%e^-(a/(2*p))*(-1)^(-v-1/2)*gamma(v+u+1/2)
+	 /(sqrt(a)*p^u*gamma(2*v+1))$
+
+/*
+ * (35)
+ * t^(u-1)*bessel_j(2*v,2*sqrt(a)*sqrt(t)) ->
+ *    gamma(u+v)*a^v/gamma(2*v+1)/p^(u+v)*%f[1,1](u+v,2*v+1,-a/p)
+ */
+prefer_whittaker:false;
+false$
+(assume(v+u>0),true);
+true$
+
+specint(t^(u-1)*bessel_j(2*v,2*sqrt(a)*sqrt(t))*exp(-p*t),t);
+a^v*p^(-v-u)*gamma(v+u)*%f[1,1]([v+u],[2*v+1],-a/p)/gamma(2*v+1)$
+
+/*
+ * (45)
+ * bessel_y(v,a*t) ->
+ *    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))$