A Nicholson-type integral for the cross-product of the Bessel functions

Hrycak, Tomasz; Schmutzhard, Sebastian

doi:10.1016/j.jmaa.2015.11.056

Cited by 4 publications

(2 citation statements)

References 11 publications

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“…Moreover, we deduce Finally, splitting the integrand into a sum of the odd and even parts we write Equation (9) in the simpler form. The combination of Equations (9) and (10) gives the desired auxiliary integral…”

Section: Integral Representation Of I µ I νmentioning

confidence: 99%

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Integral Representations for Products of Two Bessel or Modified Bessel Functions

Maširević

Pogány

2019

Mathematics

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The first part of the article contains integral expressions for products of two Bessel functions of the first kind having either different integer orders or different arguments. A similar question for a product of modified Bessel functions of the first kind is solved next, when the input functions are of different integer orders and have different arguments. Keywords: Bessel function of the first kind; modified Bessel function of the first kind; integral representation of product functions; Chebyshev polynomial of the first kind MSC: Primary: 33C10; Secondary: 33C05; 33C20 √ X 2 e it + x 2 e −it , while Nicholson proved that ([5], p. 236, Equation (39)) J n (z)J −n (z) = 1 π sin(nπ) π 0 sin(2nx) J 0 (2z sin x) dx, where z ∈ R and n / ∈ Z. This result implies for n non-negative integer the formula ([5], p. 236, Equation (40)) J 2 n (z) = 1 π π 0 sin(2nx) J 0 (2z sin x) dx .Nicholson also derived integral representation for the mixed product J n (z)Y n (z) of the Bessel function of the first kind J n (z) and of the Bessel function of the second kind Y n (z) when n ∈ Z; similar results can be found also in the already cited papers by Dixon and Ferrar [4,6] and by Görlich et al. [7].

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Section: Integral Representation Of I µ I νmentioning

confidence: 99%

“…Finally, Hrycak and Schmutzhard ( [9], p. 171) studied an integral representation for a product of two modified Bessel functions of the second kind K ν K µ and established integral representation for the cross-product of the Bessel functions J ν and Y µ , which is a generalization of Nicholson's integral for…”

Section: Introductionmentioning

confidence: 99%