rtoy
commented
4 months ago Imported from SourceForge on 2024-07-04 21:17:31 Created by crategus on 2009-09-19 16:36:31 Original: https://sourceforge.net/p/maxima/bugs/1769/#100b
I had a look into the code. We do the following transformation for a negative second parameter m:
assoc_legendre_p(n,-m,x) -> factorial(n+m)/factorial(n-m) * asscoc_legendre_p(n,m,x)
That is the simplified formula A&S 8.2.5 for m an integer.
Wolfram functions gives an additional factor (-1)^m. I think wolfram function is correct. With this factor we would get the expected results for the Laplace transform of t^n*bessel_j(1,t) too.
Furthermore we would get the correct result for the Laplace transform of t^u*bessel_j(v,z). The expected answer of example 73 in rtest14.mac is not correct. We can see it when we insert specific values in the result.
Dieter Kaiser
Imported from SourceForge on 2024-07-04 21:17:30 Created by crategus on 2009-09-19 15:15:55 Original: https://sourceforge.net/p/maxima/bugs/1769
The sign of assoc_legendre_p(n,-1,x) is wrong. We get:
(%i32) assoc_legendre_p(1,-1,x); (%o32) -sqrt(1-x^2)/2
(%i33) assoc_legendre_p(2,-1,x); (%o33) -x*sqrt(1-x^2)/2
(%i36) factor(ratsimp(assoc_legendre_p(3,-1,x))); (%o36) -sqrt(1-x^2)*(5*x^2-1)/8
(%i37) factor(ratsimp(assoc_legendre_p(4,-1,x))); (%o37) -x*sqrt(1-x^2)*(7*x^2-3)/8
In all cases above we obtain the expected answer when we multiply the result of Maxima with -1. As a reference I have taken the results from wolfram alpha.
This bug is related to the problem that
specint(exp(-s*t)*t^n*bessel_j(1,t),t)
does not work for a general power n.
Dieter Kaiser