Integrals involving products of Airy functions, their derivatives and Bessel functions

A new integral representation of the Hankel transform type is deduced for the function Fn(x,Z)=Zn−1Ai(x−Z)Ai(x+Z) with x∈R, Z>0 and n∈N. This formula involves the product of Airy functions, their derivatives and Bessel functions. The presence of the latter allows one to perform various transformations with respect to Z and obtain new integral formulae of the type of the Mellin transform, K-transform, Laplace and Fourier transform. Some integrals containing Airy functions, their derivatives and Chebyshev polynomials of the first and second kind are computed explicitly. A new representation is given for the function ²|Ai(z)| with z∈C.     

Integrals involving Whittaker functions

Summary Integrals involving products of two Whittaker functions and Bessel functions are evaluated in §§ 3, 4. Also the integrals are evaluated in § 5 while in § 6 integrals involving the product of three Whittaker functions are established.     

Some expansion formulae for G-function involving bessel functions

In this paper we have derived some expansions for G-function involving Bessel functions with the help of the orthogonality property of Bessel functions.     

On dual and triple integral equations involving modified bessel functions

A generalisation of some tripe integral equations occuring in the solution of certain mixed boundary value problems involving the wave equation is investigated. It is found that the solution of the equations can be expressed in terms of the solution of a Fredholm intgral equation of the second kind and that a related pair of dual integral equations can be solved exactly     

Bounds for modified bessel functions

Two-side inequalities for the modified Bessel functionI _v(x), K_v(x) of the first and third kind and of order v, are established. The chief tool is the monotonocity of the functionsI _v+1(x)/I _v(x),K _v+1(x)/K _v(x).     

Integrals of products of Bessel functions for applied electrodynamic problems

We determine definite integrals of products of Bessel functions of the first and second kind of different orders with various complicated arguments with finite limits of integration.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 566–570, April, 1995.     

A monotonicity property of bessel functions

Some monotonicity results are given for the remainder terms in the asymptotic expansion for x → ∞ of the function J_v(x)J_v+n(x) + Y_v(x Y_v+n(x), v ε R, n ε Z.     

Two definite integrals involving products of four Legendre functions

The definite integrals \( \int _{-1}^1x[P_\nu (x)]^4\mathrm{d}x\) and \( \int _{0}^1x[P_\nu (x)]^2\{[P_\nu (x)]^2-[P_\nu (-x)]^2\}\mathrm{d}x\) are evaluated in closed form, where \( P_\nu \) stands for the Legendre function of degree \( \nu \in \mathbb C\). Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green’s functions and Epstein zeta functions.     

Integrals involving complete elliptic integrals

We give a closed-form evaluation of a number of Erdélyi-Kober fractional integrals involving elliptic integrals of the first and second kind, in terms of the ₃F₂ generalized hypergeometric function. Reduction formulae for ₃F₂ enable us to simplify the solutions for a number of particular cases.     

Integrals of Bessel functions

We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique that we propose is based on the formal reduction of functions in this family to Gaussians.