On some integrals involving functions φ(x) such that φ(1/x) =√x φ(x)

One of the standard Mellin transform expressions for the Riemann zeta function ζ(s) in the critical strip 0 < Re(s) < 1, involves √ a function φ(x) which satisfies the functional equation φ(1/x) = x φ(x), and this relation gives rise to the well-known functional equation for ζ(s). Recently, the author has proposed three approximations for φ(x), all of which satisfy the same functional equation, and all give rise to Re(s) = 1/2 as a necessary and sufficient condition for the vanishing of the imaginary part of the corresponding Mellin transform expression. Accordingly, there is considerable interest in investigating various integrals involving arbitrary φ(x), and assuming only that φ(x) is any continuous function satisfying φ(1/x) = √x φ(x), and that certain infinite integrals converge. We first establish that the Laplace transform φˆ(p) of φ(x) satisfies a certain linear integral equation, and we confirm that four known functions satisfying the functional equation are indeed solutions of the integral equation.We then introduce a wider class of integrals involving φ(x) and denoted here by Z(ν,p), and we establish a simple integral identity, involving an integral of the Bessel function Jν (z) and the Laplace transform of φ(x). The special case corresponding to ν = 1/2 yields the previously mentioned integral equation for φˆ(p). We also establish that Z(ν,p) itself satisfies a certain linear integral equation, and a specific example, originally used by Polya, is given of one particular solution of the integral equation, which may be confirmed independently. One important consequence of these results is that we are able to deduce certain infinite integrals Ψν(x) involving φ(x), which satisfy the functional equation Ψν (1/x) = x2ν+1/2Ψν(x), and therefore in particular Ψ0(x) satisfies the same functional equation as φ(x). Various generalizations of Ψν(x) are presented which can be verified independently, and which apply to all values of ν for which the integrals are convergent