Evaluation of the integral $\smallint ^{\infty }_{0}t^{2^{\alpha }-1}J\nu (\chi \surd (1+t^{2}))/(1+t^{2})^{\alpha +\beta -1}dt$

Abstract:

Methods are developed for evaluating the integral \[ I_\nu ^{\alpha \beta }(x) = \int _0^\infty {\frac {{{t^{2\alpha - 1}}{J_\nu }(x\sqrt {1 + {t^2}} )}}{{{{(1 + {t^2})}^{\alpha + \beta - 1}}}}dt,} \] where ${J_\nu }(t)$ is the Bessel function of the first kind and order $\nu$, $\alpha > 0$, $\beta > 1/4$, and $\nu$ is real. Only $I_\nu ^{1/2,1}(x)$ and $I_\nu ^{\alpha ,\nu /2 + 1 - \alpha }(x)$ are included in previously published tables of integrals of Bessel functions. The integrals $I_1^{1/2,1/2}(x)$ and $I_2^{1/2,1}(x)$ are used in a technique developed by I. S. Fedorova for calculating the diameter distribution of long circular cylinders from small-angle x-ray, light, or neutron scattering data. The $I_\nu ^{\alpha \beta }(x)$ are shown to be proportional to a G function. From this result, power series expansions and recurrence relations are developed for use in evaluating the $I_\nu ^{\alpha \beta }(x)$. A convenient expression is obtained for the quantity required in Fedorova’s method for computing diameter distributions.