An closed-form expression of an integral of Chebyshev series and exponential function
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Does the following integral has a closed-form expression?
$int_{-1}^{1} T_n(x)exp(ipi x)dx,$
where, $T_n$ is the Chebyshev polynomial of degree n.
integration polynomials orthogonal-polynomials
asked Nov 16 at 20:52
Ray
767
add a comment |
up vote
2
down vote
favorite
Does the following integral has a closed-form expression?
$int_{-1}^{1} T_n(x)exp(ipi x)dx,$
where, $T_n$ is the Chebyshev polynomial of degree n.
integration polynomials orthogonal-polynomials
asked Nov 16 at 20:52
Ray
767
Have you tried using the explicit form $T_n(x)=cos (n arccos x)$? Maybe some Bessel functions would appear?
– Yuriy S
Nov 16 at 21:08
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Does the following integral has a closed-form expression?
$int_{-1}^{1} T_n(x)exp(ipi x)dx,$
where, $T_n$ is the Chebyshev polynomial of degree n.
integration polynomials orthogonal-polynomials
asked Nov 16 at 20:52
Ray
767
Does the following integral has a closed-form expression?
$int_{-1}^{1} T_n(x)exp(ipi x)dx,$
where, $T_n$ is the Chebyshev polynomial of degree n.
integration polynomials orthogonal-polynomials
integration polynomials orthogonal-polynomials
asked Nov 16 at 20:52
Ray
767
asked Nov 16 at 20:52
Ray
767
asked Nov 16 at 20:52
Ray
767
asked Nov 16 at 20:52
Ray
767
asked Nov 16 at 20:52
Ray
767
767
Have you tried using the explicit form $T_n(x)=cos (n arccos x)$? Maybe some Bessel functions would appear?
– Yuriy S
Nov 16 at 21:08
add a comment |
Have you tried using the explicit form $T_n(x)=cos (n arccos x)$? Maybe some Bessel functions would appear?
– Yuriy S
Nov 16 at 21:08
Have you tried using the explicit form $T_n(x)=cos (n arccos x)$? Maybe some Bessel functions would appear?
– Yuriy S
Nov 16 at 21:08
Have you tried using the explicit form $T_n(x)=cos (n arccos x)$? Maybe some Bessel functions would appear?
– Yuriy S
Nov 16 at 21:08
add a comment |
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Have you tried using the explicit form $T_n(x)=cos (n arccos x)$? Maybe some Bessel functions would appear?
– Yuriy S
Nov 16 at 21:08