Applications of Analytic Continuation to Tables of Integral Transforms and Some Integral Equations with Hyper-Singular Kernels

Analytic continuation of some classical formulas with respect to a parameter is discussed. Examples are presented. The validity of these formulas is greatly expanded. Application of these results to solving some integral equations with hyper-singular kernels is given.


Introduction
In [1] one finds several formulas of integral transforms the validity of which can be greatly expanded by analytic continuation with respect to a parameter. This is of interest per se, but also is important in applications. Analytic continuation with respect to the parameter can be used in a study of integral equations with hyper-singular kernels. This is done in Section 3. The examples of the integral equations are chosen to demonstrate that some integral equations, which do not make sense classically (that is, from the classical point of view), can be understood using the analytic continuation. Moreover, they can be solved analytically Example 1. In [1] Formula (1) in Section 2.3. is given in the form: Here and below ( ) ν Γ is the Gamma function. Classically the integral on the left in (1) diverges if Re 0 ν < or Re 2 ν > . On the other hand, in many applications, one has to consider ν outside the region specified in (1). The right side of Formula (1) admits analytic continuation with respect to ν . Indeed, if 0 y > , which we assume throughout, then is an analytic function of z on the complex plane z except for a discreet set of points 0, 1, 2, z = − −  , at which it has simple poles with known residues, see [5]. Therefore ( )

More Examples
In [1], Formula (4) in Section 2.5. is: We leave it for the reader to discuss the analytic continuation of Formulas (2) and (3) with respect to ν .

Some Applications
Consider an integral equation Our goal is to give sense to this equation and solve it analytically. One knows In the region Re 0 λ > one can take the Laplace transform of Equation (5) classically and get The question is: under what conditions the right side of Formula (8) is the Laplace transform of a function q from some functional class? The answer to this question depends on λ and h. Equation (7) makes sense by an analytic continuation with respect to λ for λ ∈  except for the points 0, 1, 2, λ = − −  . Therefore, Equation (4) can be considered as a particular case of Equation (5) with 1 4 λ = − . At this value of λ this equation is well defined by analytic continuation, although classically its kernel is hyper-singular and the integral in (4) diverges classically.
To solve Equation (4), we apply the Laplace transform and continue analytically the result with respect to λ , as was explained above. For 1 4 λ = − Equation (7) yields: One has ( ) ( ) Consider the function This is the inverse of the Laplace transform of ( ) The integral (13) converges absolutely under our assumptions since the integrand is One checks that The new feature, compared with Theorem 2, is the existence of the singularity We assume for simplicity that ( ) ( ) h t C + ∈  has compact support.
In this case ( ) L h is an entire function of p and the behavior for large t of the solution ( ) q t , found in Theorem 3 (see below), is easy to estimate.
Let us investigate the function One has: Therefore, taking 2 a β − = , one derives: e . We have proved the following theorem.
Theorem 3. Assume that h is compactly supported. Then Equation (5) with 1 2 λ = − is uniquely solvable and its solution is given by Formula (25), where ( ) g t is given by Formula (24).
The behavior of the solution for t → ∞ depends on λ , on the sign in front of the integral Equation (5)

Conclusion
It is proved in this paper that the validity of some formulas in the tables of integral transforms can be greatly expanded by the analytic continuation with respect to parameters. This idea is used for the investigation of some integral equations with hyper-singular kernels. Such an equation, see (4), plays a crucial role in the author's investigation of the Navier-Stokes problem, see [2] [3] [4].

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.