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A survey of numerical methods for the computation of Bessel function integrals. (English) Zbl 0598.65008
Rend. Sem. Mat., Torino, Fasc. Spec., Special functions: theory and computation, Int. Conf., Turin 1984, 249-265 (1985).

[For the entire collection see Zbl 0585.00004.]

The authors discuss methods for computing numerically the integral $I\left(a,p,\nu \right)={\int }_{0}^{a}{J}_{\nu }\left(px\right)f\left(x\right)dx$ where a is positive real, ${J}_{\nu }\left(·\right)$ is the Bessel function of real order $\nu$, and p is a positive parameter. These integrals are difficult to compute if ap is large, if a is infinite, or if f(x) has a singularity or is oscillatory.

Two methods are described to deal with the case where a is finite. One method subdivides the interval [0,a], integrating separately between the zeros ${j}_{\nu ,n}$ of ${J}_{\nu }\left(x\right)$, and using the Euler transformation to speed up the convergence of the resulting series. The other method is based on a modified Clenshaw-Curtis quadrature formula. Considering the integral $I={\int }_{0}^{1}{x}^{\alpha }{J}_{\nu }\left(\omega x\right)g\left(x\right)dx$ where $\alpha$ is real and such that the integral converges and g(x) is smooth, g(x) is replaced by an expansion in the shifted Chebyshev polynomials ${T}_{k}^{*}\left(x\right)$. This requires the calculation of the modified moments ${M}_{k}\left(\omega ,x,\alpha \right)={\int }_{0}^{1}{x}^{\alpha }{J}_{\nu }\left(\omega x\right){T}_{k}^{*}\left(x\right)dx·$ These moments satisfy a homogeneous linear 9-term recurrence relation. Ways to obtain starting values for this relation are indicated and its stability is analysed. It turns out that ${M}_{k}\left(\omega ,x,\alpha \right)$ can be computed accurately using forward recursion provided $k<<\omega /2$. Two examples are given and the results are compared with other methods.

In the case where the interval of intergration is infinite, four methods are briefly described. These methods, fairly wellknown to numerical analysts, consist of (i) integrating between the zeros of ${J}_{\nu }\left(x\right)$ with convergence acceleration for the resulting series, (ii) transforming the integral into a double integral by using a suitable integral representation of ${J}_{\nu }\left(x\right)$, (iii) replacing f(x) by an expansion in suitable functions and (iv), treating the tail of the integral by asymptotic methods. Howevever, it seems doubtful whether method (ii) is in fact advantageous unless analytic integration can be used to avoid numerical evaluation of the double integral. The paper closes by announcing a project for the construction of numerical software for the computation of the modified moments for 19 different weight functions.

Reviewer: K.S.Kölbig
##### MSC:
 65D20 Computation of special functions, construction of tables 65D32 Quadrature and cubature formulas (numerical methods) 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$