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On the evaluation of the integral over the product of two spherical Bessel functions. (English) Zbl 0735.33002
The integrals are of the type $I\sb{\ell,\ell'}(k,k')=\int\sb 0\sp \infty j\sb{\ell}(kr)j\sb{\ell'}(k'r)r\sp 2dr$, where $j\sb k(z)$ denotes the spherical Bessel function. As the author observes these are special cases of the well-studied Weber-Schafheitlin integrals, and which in case of convergence can be expressed in terms of the Gaussian hypergeometric functions. The present paper investigates the case of divergent integrals, as in the above example. The integral is expressed in terms of the Dirac delta function and the step function, combined with Legendre polynomials or Legendre functions. The cases $\ell-\ell'$ odd and $\ell- \ell'$ even give quite different results.
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
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