On the numerical evaluation of an oscillating infinite series. (English) Zbl 0684.65003
The series has the form \(\Sigma m\{m(m^ 2+u^ 2)^{-1}-1\}J_ 0(am)J_ 0(bm),\) where u, a, b are constants. This is not suitable for direct numerical evaluation because of the oscillation of the Bessel functions. It is transformed by contour integration and the ingenious use of Bessel function identities into a tractable form. Some results for special cases are presented in diagrams.
Reviewer: R.P.Boas
MSC:
| 65B10 | Numerical summation of series |
| 40A30 | Convergence and divergence of series and sequences of functions |
References:
| [1] | Dragoş, L., Magnetofluid Dynamics (1975), Abacus Press: Abacus Press Tunbridge Wells |
| [2] | Eason, G.; Nobel, B.; Sneddon, I. N., On certain integrals of Lipschitz-Hankel type involving products of Bessel functions, Philos. Trans. Roy. Soc. London Ser. A, 247, 529-551 (1955) · Zbl 0064.06503 |
| [3] | Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals Series and Products (1965), Academic Press: Academic Press New York · Zbl 0918.65002 |
| [4] | Sneddon, I. N., Mix Boundary Value Problems in Potential Theory (1966), North-Holland: North-Holland New York · Zbl 0149.22301 |
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