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Rayleigh Function


The Rayleigh functions sigma_n(nu) for n=1, 2, ..., are defined as

 sigma_n(nu)=sum_(k=1)^inftyj_(nu,k)^(-2n),

where +/-j_(nu,k) are the zeros of the Bessel function of the first kind J_nu(z) (Watson 1966, p. 502; Gupta and Muldoon 1999). They were used by Euler, Rayleigh, and others to evaluate zeros of Bessel functions.

There is a convolution formula connecting Rayleigh functions of different orders,

 sigma_n(nu)=1/(nu+n)sum_(k=1)^(n-1)sigma_k(nu)sigma_(n-k)(nu)

(Kishore 1963, Gupta and Muldoon 1999).


See also

Bessel Function of the First Kind

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References

Gupta, D. P. and Muldoon, M. E. "Riccati Equations and Convolution Formulas for Functions of Rayleigh Type." 24 Oct 1999. http://arxiv.org/abs/math.CA/9910128.Ismail, M. E. H. and Muldoon, M. E. "Bounds for the Small Real and Purely Imaginary Zeros of Bessel and Related Functions." Meth. Appl. Anal. 2, 1-21, 1995.Kishore, N. "The Rayleigh Function." Proc. Amer. Math. Soc. 14, 527-533, 1963.Obi, E. C. "The Complete Monotonicity of the Rayleigh Function." J. Math. Anal. Appl. 77, 465-468, 1980.Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.

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Rayleigh Function

Cite this as:

Weisstein, Eric W. "Rayleigh Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RayleighFunction.html

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