Rayleigh function

Input interpretation

Rayleigh function

Definition

The Rayleigh functions σ_n(ν) for n = 1, 2, ..., are defined as σ_n(ν) = sum_(k = 1)^∞ j_(ν, k)^(-2 n), where ± j_(ν, k) are the zeros of the Bessel function of the first kind J_ν(z). 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, σ_n(ν) = 1/(ν + n) sum_(k = 1)^(n - 1) σ_k(ν) σ_(n - k)(ν) (Kishore 1963, Gupta and Muldoon 1999).
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Related term

Bessel function of the first kind

Subject classifications

MathWorld

Bessel functions | general series

MSC 2010

33C10 | 40-XX
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