The Rayleigh function

Kishore, Nand

doi:10.1090/s0002-9939-1963-0151649-2

Cited by 56 publications

(23 citation statements)

References 2 publications

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“…By j ν,n and j ν,n , n = 1, 2, ... we indicate the n-th positive zeros of J ν (x) and J ν (x) respctively. Using only these representations for J ν (x) and J ν (x) we obtain very easily well known [1,2,3,5] results concerning the zeros of these functions.…”

Section: Introductionmentioning

confidence: 79%

Simple proofs of classical results on zeros of $\mathbf{J_\nu(x)}$ and $\mathbf{J_\nu'(x)}$

Kokologiannaki

Laforgia

2014

Tbilisi Math. J.

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Section: Introductionmentioning

confidence: 79%

Simple proofs of classical results on zeros of $\mathbf{J_\nu(x)}$ and $\mathbf{J_\nu'(x)}$

Kokologiannaki

Laforgia

2014

Tbilisi Math. J.

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“…. (1.2) Later it was also established by N. Kishore [7]. Combined with the representation for the first Rayleigh sum (see [20, p. 502]),…”

Section: Introductionmentioning

confidence: 87%

“…Convolutions of Rayleigh functions with respect to the power l were considered in [1,3,4,7,10,11]. N. Meiman [10] was the first to obtain a compact recursion formula σ l (ν) = 1 ν + l l−1 k=1 σ l−k (ν)σ k (ν) for l = 2, 3, 4, .…”

Section: Introductionmentioning

confidence: 99%

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Варламов¹

2007

Journal of Mathematical Analysis and Applications

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Rayleigh functions σ l (ν) are defined as series in inverse powers of the Bessel function zeros λ ν,n = 0, σ l (ν) = ∞ n=1 1 λ 2l ν, n , where l = 1, 2, . . . ; ν is the index of the Bessel function J ν (x) and n = 1, 2, . . . is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index, R l (m) = ∞ k=−∞ σ l |m − k| σ l |k| for l = 1, 2, . . . ; m = 0, ±1, ±2, . . . , are needed for constructing global-in-time solutions of semi-linear evolution equations in circular domains [V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal. 46 (2001) 699-725; V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424]. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424], where the properties of R 1 (m) were investigated.In the present work a general representation of R l (m) in terms of σ l (ν) is deduced. On the basis of this a representation for the function R 2 (m) is obtained in terms of the ψ-function. An asymptotic expansion is computed for R 2 (m) as |m| → ∞. Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in bounded ✩ domains with periodicity conditions in one coordinate. As an example of application of R l (m) a forced Boussinesq equation u tt − 2bΔu t = −αΔ 2 u + Δu + βΔ u 2 + f with α, b = const > 0 and β = const ∈ R is considered in a unit disc with homogeneous boundary and initial data. Construction of its global-in-time solutions involves the use of the functions R 1 (m) and R 2 (m) which are responsible for the nonlinear smoothing effect.

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“…(see, e.g., [1][2][3]5,6] and the references cited therein). Therefore R(m) can be considered a new special function.…”

Section: Introductionmentioning

confidence: 99%

Convolution of Rayleigh functions with respect to the Bessel index

Варламов¹

2005

Journal of Mathematical Analysis and Applications

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A convolution of Rayleigh functions with respect to the Bessel index can be treated as a special function in its own right. It appears in constructing global-in-time solutions for some semilinear evolution equations in circular domains and may control the smoothing effect due to nonlinearity. An explicit representation for it is derived which involves the special function ψ(x) (the logarithmic derivative of the Γ -function). The properties of the convolution in question are established. Asymptotic expansions for small and large values of the argument are obtained and the graph is presented. Published by Elsevier Inc.

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The Rayleigh function

Cited by 56 publications

References 2 publications

Simple proofs of classical results on zeros of $\mathbf{J_\nu(x)}$ and $\mathbf{J_\nu'(x)}$

Simple proofs of classical results on zeros of $\mathbf{J_\nu(x)}$ and $\mathbf{J_\nu'(x)}$

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Convolution of Rayleigh functions with respect to the Bessel index

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