$\int_0^\infty dx \frac{xJ_0(ax)}{\sqrt{1+x^2}}$, where $J_0$ is the zeroth-order Bessel function.
2026-03-29 11:06:03.1774782363
Integral involving Bessel function
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This is called the Hankel transform and corresponds to the Fourier en transform in dimension $2$ of $\frac{1}{\sqrt{1+|x|^2}}$ (see Fourier transform of a radial function or Equation (23)). As written Here, it is $$\frac{e^{-a}}{a}$$