/* This is a sample of answers from the under construction program, 
which deals with "Bateman's Manuscript Project". */

assume(p > 0, a > 0, b > 0);

/* laplace transforms */

/* some elementary functions. */

t^(1/2)*%e^(-a*t/4)*%e^(-p*t);
specint(%,t);

t^(3/4)*%e^(-t^2/2/b)*%e^(-p*t);
specint(%,t);

t^(-1/2)*%e^(-2*a^(1/2)*t^(1/2))*%e^(-p*t);
specint(%,t);

/* Next example triggers a bug:
 * "Is  psey  positive, negative, or zero?"
 * psey is an internal variable
 * bug report # 1635365
 *
t^(1/2)*%e^(-p*t-a/t);
specint(%,t);
 */

/* Next example triggers a bug:
 * internal variable fail-on-f24p146test appears
 * bug report # 1635372
 *
sin(a*t)*cosh(b*t^2)*%e^(-p*t);
radcan(specint(%,t));
 */


/* some "confluents". notice that "%m[k,m](z)" is a whittaker function. */

%e^(-a*t)*t^2*erf(t^(1/2))*%e^(-p*t);
specint(%,t);
factor(radcan(%));

t^(3/2)*%m[1/2,1](t)*%e^(-p*t);
specint(%,t);
factor(radcan(%));


/* some bessel functs (bf's). */

/* %j[v](z), 1st kind of bf's. */
	/* %y[v](z), 2nd kind of bf's.*/
/* %h[v,1](z), 1st kind of the 3rd kind of bf's (1st hankel). */
	/* %h[v,2](z), 2nd kind of the 3rd kind of bf's (2nd hankel).*/

t^(1/2)*bessel_j(1, 2*a^(1/2)*t^(1/2))*%e^(-p*t);
specint(%,t);

t^2*bessel_j(1, a*t)*%e^(-p*t);
specint(%,t);
factor(radcan(%));

t^(3/2)*bessel_y(1, a*t)*%e^(-t);
specint(%,t);

t^(3/2)*hankel_1(1/2,t)*%e^(-p*t);
specint(%,t);
factor(radcan(%));

t^(1/2)*hankel_2(3/4,t)*%e^(-p*t);
specint(%,t);
ratsimp(%);

t*hankel_1(2/3,t^(1/2))*%e^(-p*t);
specint(%,t);
ratsimp(%);


/* bessel_i(v,z), %k[v](z), modified bf's. */

t^(1/2)*bessel_i(1,t)*%e^(-p*t);
specint(%,t);
ratsimp(%);

bessel_j(1, t)^2*%e^(-p*t+r);
specint(%,t);

t^(1/2)*bessel_j(1/2, t^(1/2))^2*%e^(-p*t);
specint(%,t);

t^(5/2)*bessel_y(1/2, t^(1/2))^2*%e^(-p*t);
specint(%,t);
ratsimp(%);

/*some combinations of products of bessel functions*/
t^(3/4)*bessel_j(1/2, t)*bessel_j(1/4, t)*%e^(-p*t);
specint(%,t);
ratsimp(%);

/* Next example triggers asksign of (psey - a)*(psey + a)
 * bug report # 1635365
 *
t*bessel_i(0,a*t/2)*bessel_i(1,a*t/2)*%e^(-p*t);
specint(%,t);
 */


/* related to bf's functions. */

/* struve functions. */

t^(3/2)*struve_h(1,t^(1/2))*%e^(-p*t);
specint(%,t);
radcan(%);

t*struve_h(1,t)*%e^(-p*t);
specint(%,t);
radcan(%);

forget(p > 0, a > 0, b > 0);
