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Register a new userIntegral formula for the Bessel function of the first kind
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Authors
Enrico De Micheli
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In this paper, we prove a new integral representation for the Bessel function of the first kind $J_mu(z)$, which holds for any $mu,zinmathbb{C}$.
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A Fourier-type integral representation for Bessels function of the first kind and complex order is obtained by using the Gegenbuaer extension of Poissons integral representation for the Bessel function along with a trigonometric integral representation of Gegenbauers polynomials. This representation lets us express various functions related to the incomplete gamma function in series of Bessels functions. Neumann series of Bessel functions are also considered and a new closed-form integral representation for this class of series is given. The density function of this representation is simply the analytic function on the unit circle associated with the sequence of coefficients of the Neumann series. Examples of new closed-form integral representations of special functions are also presented.
In this paper, sums represented in (3) are studied. The expressions are derived in terms of Bessel functions of the first and second kinds and their integrals. Further, we point out the integrals can be written as a Meijer G function.
This note concerns the general formulation by Preiss and Uher of Kestelmans influential result pertaining the change of variable, or substitution, formula for the Riemann integral.
We consider the Steklov zeta function $zeta$ $Omega$ of a smooth bounded simply connected planar domain $Omega$ $subset$ R 2 of perimeter 2$pi$. We provide a first variation formula for $zeta$ $Omega$ under a smooth deformation of the domain. On the base of the formula, we prove that, for every s $in$ (--1, 0) $cup$ (0, 1), the difference $zeta$ $Omega$ (s) -- 2$zeta$ R (s) is non-negative and is equal to zero if and only if $Omega$ is a round disk ($zeta$ R is the classical Riemann zeta function). Our approach gives also an alternative proof of the inequality $zeta$ $Omega$ (s) -- 2$zeta$ R (s) $ge$ 0 for s $in$ (--$infty$, --1] $cup$ (1, $infty$); the latter fact was proved in our previous paper [2018] in a different way. We also provide an alternative proof of the equality $zeta$ $Omega$ (0) = 2$zeta$ R (0) obtained by Edward and Wu [1991].
A new recurrence relation for exceptional orthogonal polynomials is proposed, which holds for type 1, 2 and 3. As concrete examples, the recurrence relations are given for Xj-Hermite, Laguerre and Jacobi polynomials in j = 1,2 case.
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