custom Maple Legendre functions

Custom Maple Legendre functions.doc PhL 2.8.11 I keep having problems with evaluation of LegendreP and LegendreQ failing, but I think the hypergeom function does not have any such problems. So I will try to just write my own functions customized to what I am looking for. 1. Mehler functions. In my doc "doing Mehler functions" Appendix A I claim that P-1/2+iτ (chα) = e-(1/2)αeiατ F(1/2-iτ, 1/2; 1; 1 - e-2α) Let's first try this in Maple this way] Pmehler := (alpha,tau) -> exp(-alpha/2 + I*alpha*tau)*hypergeom([1/2-I*tau, 1/2],[1], 1 - exp(-2*alpha)); We always need to "do a test" to make sure things got entered correctly, so and I am happy to have that tiny imaginary part which can be removed. Now we try a simple plot and this is what I was wanting to see. But the LegendreP plots works fine, I am unable to replicate the problem I was having! 2. Toroidal functions. OK, I will put this doc on hold for a while until the problem arises again. // That didn't take long. Here is a problem: Maple refuses to compute this function. So let's try to make our own version. Ptoroidal(nu,alpha). We now need a Q rep that converges on the positive real axis above 1. Looking at the converge doc, there are two choices: (42) and (46). They both have two terms, fine. arg for (42) is 1-1/z2 and with z = chξ this would be 1-sech2ξ = th2ξ so that seems pretty good. So here is the entry Qtor := (xi,nu) -> stop. It has Γ(μ) sitting there, no good since μ = 0. How about (46)? Same problem. What does this mean? How about (36) which should work at least down to 1.00001. arg for (36) = 2/(1+z) = 2/(1+chξ) = 1/ ch2(ξ/2) which is again fine. And (36) has only one term. So we try f1 := 2^(nu); f2 := (GAMMA(1+nu))^2; f3 := (cosh(xi)+1)^(-nu-1)/GAMMA(2+2*nu); f4 := hypergeom([1+nu,1+nu],[2+2*nu],1/(cosh(xi/2))^2; Qtor :=(xi,nu) -> 2^(nu)*(GAMMA(1+nu))^2*(cosh(xi)+1) OK, here now is a specific example of the problem: My call to Qtor(-1/2,a) gives a result, but the same LegendreQ refuses to evaluate. It is not a cut problem because this also fails Now having shown up a problem, let's show that we agree at some other value: Here LegendreQ seems able to evaluate, and we get the same result. So we seem pretty successful with (36) for Q. The "partner": to (36) is (18) but that has two terms. Can we use the same P as used above, which was this P-1/2+iτ (chα) = e-(1/2)αeiατ F(1/2-iτ, 1/2; 1; 1 - e-2α) Pν(z) = zν F(-ν/2, 1/2-ν/2; 1; 1-1/z2) So we could define Ptor(nu,xi) Pν(z) = zν F(-ν/2, 1/2-ν/2; 1; 1-1/z2) P(ν,ξ) ≡ Pν(chξ) = (chξ)ν F(-ν/2, 1/2-ν/2; 1; th2ξ) Qν(z) = 2ν [Γ(1+ν)]2 (z+1)-ν-1/Γ(2+2ν) * F(1+ν,1+ν,2+2ν, 2/(1+z) Q(ν,ξ) = Qν(chξ) = 2ν [Γ(1+ν)]2 (chξ+1)-ν-1/Γ(2+2ν) * F(1+ν,1+ν,2+2ν, 2/(1+chξ) I have now entered the above stuff into PandQ.mws and it looks very good Then I can construct the kinds of sums we get in donut potential!