ahmadtalaei

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These are replies submitted by ahmadtalaei

@ahmadtalaei 

 

Thanks a lot for your help.

@dharr 

 

Thank you so much. Mine is 2016 and that is probably the issue. Anyway, could you please convert two hypergeom functions to given in the attached file to the LagendreP?

restart

sol[1] := hypergeom([3/4+(1/4)*sqrt(1+4*_c[1]), 3/4-(1/4)*sqrt(1+4*_c[1])], [1/2], cos(phi)^2)+hypergeom([5/4+(1/4)*sqrt(1+4*_c[1]), 5/4-(1/4)*sqrt(1+4*_c[1])], [3/2], cos(phi)^2);

hypergeom([3/4+(1/4)*(1+4*_c[1])^(1/2), 3/4-(1/4)*(1+4*_c[1])^(1/2)], [1/2], cos(phi)^2)+hypergeom([5/4+(1/4)*(1+4*_c[1])^(1/2), 5/4-(1/4)*(1+4*_c[1])^(1/2)], [3/2], cos(phi)^2)

(1)

convert(sol[1], LegendreP)

``


 

Download convert-Legendre-v1.mw


 

restart

with(PDETools):

infolevel[pdsolve] := 3:

sol[1] := dsolve((1-x^2)*(diff(y(x), x, x))+n*(n+1)*y(x) = 0)

y(x) = _C1*(1-x^2)*hypergeom([1+(1/2)*n, 1/2-(1/2)*n], [1/2], x^2)+_C2*(-x^3+x)*hypergeom([1-(1/2)*n, 3/2+(1/2)*n], [3/2], x^2)

(1)

convert(sol[1], LegendreP)

y(x) = _C1*(1-x^2)*hypergeom([1+(1/2)*n, 1/2-(1/2)*n], [1/2], x^2)+_C2*(-x^3+x)*hypergeom([1-(1/2)*n, 3/2+(1/2)*n], [3/2], x^2)

(2)

sol[2] := simplify(convert(_C1*(1-cos(phi)^2)^(3/4)*(-cos(phi)^2)^(1/4)*sqrt(Pi)*LegendreP((1/2)*n-1/2, 1/2, 1-2*cos(phi)^2), hypergeom));

_C1*sin(phi)^2*hypergeom([1/2+(1/2)*n, 1/2-(1/2)*n], [1/2], cos(phi)^2)

(3)

``

(4)


 

Download convert-Legendre-v1.mw

@dharr 

I was able to upload the file. Thank you.

I find on the following page where these two functions can be converted to each other. Please see equation 14.3.1 in the following page:

https://dlmf.nist.gov/14.3

But the problem is that I couldn't find value for ν in the hypergeom function F(ν+1,−ν;1−μ;1/2−1/2x) that could result in the Legendre(ν,μ,x) where x=cos($\psi$).

@Mariusz Iwaniuk 

I always get a problem copy&paste from maple. Here is the link provided to the question:

https://math.stackexchange.com/questions/3254765/how-to-convert-a-hypergeom-function-to-the-legendre-function

@Thomas Richard 

 

Thank you for your comment.

Could you please follow the answer by @tomleslie in below which attaches a Maple file and help me to find a way to solve this PDE?

@tomleslie Thank you.

 

I thought the separation of variables could help. But do you have any suggestion on how to solve the PDE?

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