2005
DOI: 10.1016/j.jmaa.2004.12.012
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On integral representation of Bessel function of the first kind

Abstract: A first kind Fredholm integral equation with nondegenerate kernel is given, which particular solution is the Bessel function of the first kind. This equation is solved by means of Mellin transform pair.

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Cited by 8 publications
(5 citation statements)
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“…This latter relation no longer holds when n is not an integral number since, for ν ∈ Z, J ν (x) and J −ν (x) are linearly independent solutions of the Bessel equation of order ν. Numerous formulae express the Bessel functions of the first kind as definite integrals, which can be exploited to obtain, for instance, approximations and asymptotic expansions (see [27, Chapter VI], [12], and the website [11, Sect. 10.9] for a useful collection of formulae).…”
Section: Fourier-type Integral Representation Of Bessel Functions Of ...mentioning
confidence: 99%
“…This latter relation no longer holds when n is not an integral number since, for ν ∈ Z, J ν (x) and J −ν (x) are linearly independent solutions of the Bessel equation of order ν. Numerous formulae express the Bessel functions of the first kind as definite integrals, which can be exploited to obtain, for instance, approximations and asymptotic expansions (see [27, Chapter VI], [12], and the website [11, Sect. 10.9] for a useful collection of formulae).…”
Section: Fourier-type Integral Representation Of Bessel Functions Of ...mentioning
confidence: 99%
“…which consists for a polynomial of degree m in numerator, and of a polynomial of degree 2m in denominator, both in variable r 2 . By applying Mellin transform to the convolutional equation (4.1) as given in [6], we conclude…”
Section: Solving Integral Equation (31) Inmentioning
confidence: 99%
“…where a ∈ C 1 (R + ), a| N = (a n ) n∈N ; (1.8) completely solves the Open problem on the integral form of S α µ (r; a) posed by Feng Qi. Finally, let us recall in short that Draščić and Pogány [6] established a first kind Fredholm integral equation with non-degenerated kernel which connects two types of integral representations of generalized Mathieu series S µ (r). The benefit was a new integral representation for the Bessel function of the first kind with general positive order ν > 0 [6, Theorem 1, Eq.…”
Section: Introductionmentioning
confidence: 99%
“…with respect to the ordinary Lebesgue measure on the positive half-line when ∞ 0 f (x)g(x)dx vanishes, writing this as f ⊥ g. We mention that h as in the above theorem has been constructed in [16,Example]. To solve the integral equation (5.2) we use the Mellin integral transform technique, following some lines of a similar procedure used by Draščić-Pogány in [16]. The Mellin transform pair of certain suitable f we define [53] M p (f ) :=…”
Section: Differential Equations For Kapteyn and Schlömilch Series Of ...mentioning
confidence: 99%