Mixing
Finding $int_{0}^{1} frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{sqrt{x-x^2}} dx$
$begingroup$
Shifted Chebyshev polynomials $$T_{i}^{*}(x) = cos(i arccos(2x-1))$$
We want to calculate $$I=int_{0}^{1} frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{sqrt{x-x^2}} dx$$ Which is equal to $$sum_{substack{i=0 \ (i+j) even}}^{j-2} 2 delta_{ij} j(j^2-i^2)c_i$$
How to get this result
We know that
$$int_{0}^{1} frac{T_{i}^{*}(x) T_{j}^{*}(x)}{sqrt{x-x^2}} dx = delta_{ij}$$
Where $$
delta_{ij} =
left{
begin{array}{ll}
frac{pi}{c_{i}} & i=j \
0 & ineq j
end{array} , c_{i} =
left{
begin{array}{ll}
1 & i=0 \
2 & igeq 1
end{array}
right.
right.
$$
And We can write $$T_{i}^{*''}(x) =sum_{k=0}^{i-2} a_{k} T_{i}^{*}(x)$$
So how to get the required result above ?
if we got $a_{k}$ we are supposed to get the required result
How to get the required result above ?
And Is there a general formula to get $a_k$
such that $$T_{i}^{*(n)}(x) = sum_{k=0}^{i-2} a_{k} T_{i}^{*}(x) $$
Using Mathematica I have got :
integration definite-integrals orthogonal-polynomials chebyshev-polynomials
asked Jan 13 at 7:44
topspintopspin
727413
$endgroup$
add a comment |
$begingroup$
Shifted Chebyshev polynomials $$T_{i}^{*}(x) = cos(i arccos(2x-1))$$
We want to calculate $$I=int_{0}^{1} frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{sqrt{x-x^2}} dx$$ Which is equal to $$sum_{substack{i=0 \ (i+j) even}}^{j-2} 2 delta_{ij} j(j^2-i^2)c_i$$
How to get this result
We know that
$$int_{0}^{1} frac{T_{i}^{*}(x) T_{j}^{*}(x)}{sqrt{x-x^2}} dx = delta_{ij}$$
Where $$
delta_{ij} =
left{
begin{array}{ll}
frac{pi}{c_{i}} & i=j \
0 & ineq j
end{array} , c_{i} =
left{
begin{array}{ll}
1 & i=0 \
2 & igeq 1
end{array}
right.
right.
$$
And We can write $$T_{i}^{*''}(x) =sum_{k=0}^{i-2} a_{k} T_{i}^{*}(x)$$
So how to get the required result above ?
if we got $a_{k}$ we are supposed to get the required result
How to get the required result above ?
And Is there a general formula to get $a_k$
such that $$T_{i}^{*(n)}(x) = sum_{k=0}^{i-2} a_{k} T_{i}^{*}(x) $$
Using Mathematica I have got :
integration definite-integrals orthogonal-polynomials chebyshev-polynomials
asked Jan 13 at 7:44
topspintopspin
727413
$endgroup$
add a comment |
$begingroup$
Shifted Chebyshev polynomials $$T_{i}^{*}(x) = cos(i arccos(2x-1))$$
We want to calculate $$I=int_{0}^{1} frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{sqrt{x-x^2}} dx$$ Which is equal to $$sum_{substack{i=0 \ (i+j) even}}^{j-2} 2 delta_{ij} j(j^2-i^2)c_i$$
How to get this result
We know that
$$int_{0}^{1} frac{T_{i}^{*}(x) T_{j}^{*}(x)}{sqrt{x-x^2}} dx = delta_{ij}$$
Where $$
delta_{ij} =
left{
begin{array}{ll}
frac{pi}{c_{i}} & i=j \
0 & ineq j
end{array} , c_{i} =
left{
begin{array}{ll}
1 & i=0 \
2 & igeq 1
end{array}
right.
right.
$$
And We can write $$T_{i}^{*''}(x) =sum_{k=0}^{i-2} a_{k} T_{i}^{*}(x)$$
So how to get the required result above ?
if we got $a_{k}$ we are supposed to get the required result
How to get the required result above ?
And Is there a general formula to get $a_k$
such that $$T_{i}^{*(n)}(x) = sum_{k=0}^{i-2} a_{k} T_{i}^{*}(x) $$
Using Mathematica I have got :
integration definite-integrals orthogonal-polynomials chebyshev-polynomials
asked Jan 13 at 7:44
topspintopspin
727413
$endgroup$
Shifted Chebyshev polynomials $$T_{i}^{*}(x) = cos(i arccos(2x-1))$$
We want to calculate $$I=int_{0}^{1} frac{T_{i}^{*''}(x) T_{j}^{*}(x)}{sqrt{x-x^2}} dx$$ Which is equal to $$sum_{substack{i=0 \ (i+j) even}}^{j-2} 2 delta_{ij} j(j^2-i^2)c_i$$
How to get this result
We know that
$$int_{0}^{1} frac{T_{i}^{*}(x) T_{j}^{*}(x)}{sqrt{x-x^2}} dx = delta_{ij}$$
Where $$
delta_{ij} =
left{
begin{array}{ll}
frac{pi}{c_{i}} & i=j \
0 & ineq j
end{array} , c_{i} =
left{
begin{array}{ll}
1 & i=0 \
2 & igeq 1
end{array}
right.
right.
$$
And We can write $$T_{i}^{*''}(x) =sum_{k=0}^{i-2} a_{k} T_{i}^{*}(x)$$
So how to get the required result above ?
if we got $a_{k}$ we are supposed to get the required result
How to get the required result above ?
And Is there a general formula to get $a_k$
such that $$T_{i}^{*(n)}(x) = sum_{k=0}^{i-2} a_{k} T_{i}^{*}(x) $$
Using Mathematica I have got :
integration definite-integrals orthogonal-polynomials chebyshev-polynomials
integration definite-integrals orthogonal-polynomials chebyshev-polynomials
asked Jan 13 at 7:44
topspintopspin
727413
asked Jan 13 at 7:44
topspintopspin
727413
edited Jan 13 at 16:54
topspin
asked Jan 13 at 7:44
topspintopspin
727413
asked Jan 13 at 7:44
topspintopspin
727413
asked Jan 13 at 7:44
topspintopspin
727413
727413
add a comment |
add a comment |
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