Fourier-Legendre Series for $arcsin{x}$
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I have been trying to work out how to work out the coefficients in the Fourier-Legendre series of $arcsin{x}$ (i.e., find $c_n$'s s.t. $arcsin{x}=sum_{n=0}^infty c_n P_n(x)$ where $P_n(x)$ is the $n$-th Legendre Polynomial), and I have had little luck finding a general expression for the coefficients. I know they can be calculated as $$c_n=frac{int_{-1}^{1} P_n(x) arcsin{x} mathrm{d}x}{int_{-1}^{1} P_n(x)^2 mathrm{d}x}=frac{2n+1}{2}int_{-1}^{1} P_n(x) arcsin{x} mathrm{d}x$$. So far I've tried writing out $P_n$ using Rodriguez's formula ($P_ell(x)=frac{1}{2^ell ell !} frac{mathrm{d}^ell}{mathrm{d}x^ell}left[left(x^2-1right)^ellright]$) and integrating that by parts, but I'm not seeing any way to make a recurrence relation from it or get anything closer to a closed form for $c_n$. Does this seem like a problem which may simply not have a closed form answer? Any hints on how to proceed or any evidence that a closed form solution doesn't exist would be appreciated!
fourier-series legendre-polynomials
asked Nov 13 at 18:12
JoDraX
21628
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I have been trying to work out how to work out the coefficients in the Fourier-Legendre series of $arcsin{x}$ (i.e., find $c_n$'s s.t. $arcsin{x}=sum_{n=0}^infty c_n P_n(x)$ where $P_n(x)$ is the $n$-th Legendre Polynomial), and I have had little luck finding a general expression for the coefficients. I know they can be calculated as $$c_n=frac{int_{-1}^{1} P_n(x) arcsin{x} mathrm{d}x}{int_{-1}^{1} P_n(x)^2 mathrm{d}x}=frac{2n+1}{2}int_{-1}^{1} P_n(x) arcsin{x} mathrm{d}x$$. So far I've tried writing out $P_n$ using Rodriguez's formula ($P_ell(x)=frac{1}{2^ell ell !} frac{mathrm{d}^ell}{mathrm{d}x^ell}left[left(x^2-1right)^ellright]$) and integrating that by parts, but I'm not seeing any way to make a recurrence relation from it or get anything closer to a closed form for $c_n$. Does this seem like a problem which may simply not have a closed form answer? Any hints on how to proceed or any evidence that a closed form solution doesn't exist would be appreciated!
fourier-series legendre-polynomials
asked Nov 13 at 18:12
JoDraX
21628
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have been trying to work out how to work out the coefficients in the Fourier-Legendre series of $arcsin{x}$ (i.e., find $c_n$'s s.t. $arcsin{x}=sum_{n=0}^infty c_n P_n(x)$ where $P_n(x)$ is the $n$-th Legendre Polynomial), and I have had little luck finding a general expression for the coefficients. I know they can be calculated as $$c_n=frac{int_{-1}^{1} P_n(x) arcsin{x} mathrm{d}x}{int_{-1}^{1} P_n(x)^2 mathrm{d}x}=frac{2n+1}{2}int_{-1}^{1} P_n(x) arcsin{x} mathrm{d}x$$. So far I've tried writing out $P_n$ using Rodriguez's formula ($P_ell(x)=frac{1}{2^ell ell !} frac{mathrm{d}^ell}{mathrm{d}x^ell}left[left(x^2-1right)^ellright]$) and integrating that by parts, but I'm not seeing any way to make a recurrence relation from it or get anything closer to a closed form for $c_n$. Does this seem like a problem which may simply not have a closed form answer? Any hints on how to proceed or any evidence that a closed form solution doesn't exist would be appreciated!
fourier-series legendre-polynomials
asked Nov 13 at 18:12
JoDraX
21628
I have been trying to work out how to work out the coefficients in the Fourier-Legendre series of $arcsin{x}$ (i.e., find $c_n$'s s.t. $arcsin{x}=sum_{n=0}^infty c_n P_n(x)$ where $P_n(x)$ is the $n$-th Legendre Polynomial), and I have had little luck finding a general expression for the coefficients. I know they can be calculated as $$c_n=frac{int_{-1}^{1} P_n(x) arcsin{x} mathrm{d}x}{int_{-1}^{1} P_n(x)^2 mathrm{d}x}=frac{2n+1}{2}int_{-1}^{1} P_n(x) arcsin{x} mathrm{d}x$$. So far I've tried writing out $P_n$ using Rodriguez's formula ($P_ell(x)=frac{1}{2^ell ell !} frac{mathrm{d}^ell}{mathrm{d}x^ell}left[left(x^2-1right)^ellright]$) and integrating that by parts, but I'm not seeing any way to make a recurrence relation from it or get anything closer to a closed form for $c_n$. Does this seem like a problem which may simply not have a closed form answer? Any hints on how to proceed or any evidence that a closed form solution doesn't exist would be appreciated!
fourier-series legendre-polynomials
fourier-series legendre-polynomials
asked Nov 13 at 18:12
JoDraX
21628
asked Nov 13 at 18:12
JoDraX
21628
asked Nov 13 at 18:12
JoDraX
21628
asked Nov 13 at 18:12
JoDraX
21628
asked Nov 13 at 18:12
JoDraX
21628
21628
add a comment |
add a comment |
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