Bessel integral invovling algebraic and hyperbolic functions
$begingroup$
I am desperate in evaluating the following Hankel transform
$$
int_{0}^{infty} frac{J_0(kr)}{k^2+xi^2} frac{cosh(ky)}{cosh(k)} kmathrm{d} k,
$$
where $J_0(kr)$ is the Bessel function of the first kind and where the interest is focused on where $k$ is small.
By expanding the hyperbolic function about $k=0$,
$$
frac{cosh(ky)}{cosh(k)} = 1 + textit{O}(k^2),
$$
and the preceding integral reduces to
$$
int_{0}^{infty} frac{J_0(kr)}{k^2+xi^2} kmathrm{d} k = K_0(xi r),
$$
where $K_0(xi r)$ is the modified Bessel function of the second kind.
The problem is the approximation for the hyperbolic function is valid only about $k=0$, while the integral is over the $kin(0,infty)$. How can I justify this controversy?
Or, perhaps it is acceptable to evaluate
$$
int_{0}^{1} frac{J_0(kr)}{k^2+xi^2} kmathrm{d} k,
$$
instead, but how to do this?
Thanking you and, please, could you help me...
Wang Zhe
definite-integrals bessel-functions integral-transforms
asked Dec 17 '18 at 10:20
Zhe WangZhe Wang
112
$endgroup$
add a comment |
$begingroup$
I am desperate in evaluating the following Hankel transform
$$
int_{0}^{infty} frac{J_0(kr)}{k^2+xi^2} frac{cosh(ky)}{cosh(k)} kmathrm{d} k,
$$
where $J_0(kr)$ is the Bessel function of the first kind and where the interest is focused on where $k$ is small.
By expanding the hyperbolic function about $k=0$,
$$
frac{cosh(ky)}{cosh(k)} = 1 + textit{O}(k^2),
$$
and the preceding integral reduces to
$$
int_{0}^{infty} frac{J_0(kr)}{k^2+xi^2} kmathrm{d} k = K_0(xi r),
$$
where $K_0(xi r)$ is the modified Bessel function of the second kind.
The problem is the approximation for the hyperbolic function is valid only about $k=0$, while the integral is over the $kin(0,infty)$. How can I justify this controversy?
Or, perhaps it is acceptable to evaluate
$$
int_{0}^{1} frac{J_0(kr)}{k^2+xi^2} kmathrm{d} k,
$$
instead, but how to do this?
Thanking you and, please, could you help me...
Wang Zhe
definite-integrals bessel-functions integral-transforms
asked Dec 17 '18 at 10:20
Zhe WangZhe Wang
112
$endgroup$
add a comment |
$begingroup$
I am desperate in evaluating the following Hankel transform
$$
int_{0}^{infty} frac{J_0(kr)}{k^2+xi^2} frac{cosh(ky)}{cosh(k)} kmathrm{d} k,
$$
where $J_0(kr)$ is the Bessel function of the first kind and where the interest is focused on where $k$ is small.
By expanding the hyperbolic function about $k=0$,
$$
frac{cosh(ky)}{cosh(k)} = 1 + textit{O}(k^2),
$$
and the preceding integral reduces to
$$
int_{0}^{infty} frac{J_0(kr)}{k^2+xi^2} kmathrm{d} k = K_0(xi r),
$$
where $K_0(xi r)$ is the modified Bessel function of the second kind.
The problem is the approximation for the hyperbolic function is valid only about $k=0$, while the integral is over the $kin(0,infty)$. How can I justify this controversy?
Or, perhaps it is acceptable to evaluate
$$
int_{0}^{1} frac{J_0(kr)}{k^2+xi^2} kmathrm{d} k,
$$
instead, but how to do this?
Thanking you and, please, could you help me...
Wang Zhe
definite-integrals bessel-functions integral-transforms
asked Dec 17 '18 at 10:20
Zhe WangZhe Wang
112
$endgroup$
I am desperate in evaluating the following Hankel transform
$$
int_{0}^{infty} frac{J_0(kr)}{k^2+xi^2} frac{cosh(ky)}{cosh(k)} kmathrm{d} k,
$$
where $J_0(kr)$ is the Bessel function of the first kind and where the interest is focused on where $k$ is small.
By expanding the hyperbolic function about $k=0$,
$$
frac{cosh(ky)}{cosh(k)} = 1 + textit{O}(k^2),
$$
and the preceding integral reduces to
$$
int_{0}^{infty} frac{J_0(kr)}{k^2+xi^2} kmathrm{d} k = K_0(xi r),
$$
where $K_0(xi r)$ is the modified Bessel function of the second kind.
The problem is the approximation for the hyperbolic function is valid only about $k=0$, while the integral is over the $kin(0,infty)$. How can I justify this controversy?
Or, perhaps it is acceptable to evaluate
$$
int_{0}^{1} frac{J_0(kr)}{k^2+xi^2} kmathrm{d} k,
$$
instead, but how to do this?
Thanking you and, please, could you help me...
Wang Zhe
definite-integrals bessel-functions integral-transforms
definite-integrals bessel-functions integral-transforms
asked Dec 17 '18 at 10:20
Zhe WangZhe Wang
112
asked Dec 17 '18 at 10:20
Zhe WangZhe Wang
112
edited Dec 17 '18 at 15:33
Zhe Wang
asked Dec 17 '18 at 10:20
Zhe WangZhe Wang
112
asked Dec 17 '18 at 10:20
Zhe WangZhe Wang
112
asked Dec 17 '18 at 10:20
Zhe WangZhe Wang
112
112
add a comment |
add a comment |
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