I am desperate in evaluating the following Hankel transform $$ \int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} \frac{\cosh(ky)}{\cosh(k)} k\mathrm{d} k, $$ 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 $$ \int_{0}^{\infty} \frac{J_0(kr)}{k^2+\xi^2} k\mathrm{d} k = K_0(\xi r), $$ 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 $k\in(0,\infty)$. How can I justify this controversy?

Or, perhaps it is acceptable to evaluate $$ \int_{0}^{1} \frac{J_0(kr)}{k^2+\xi^2} k\mathrm{d} k, $$ instead, but how to do this?

Thanking you and, please, could you help me...

Wang Zhe


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