Abstract:
Integral equations for the spin-weighted spheroidal wave functions is given. For the prolate spheroidal wave function with m=0, there exists the integral equation whose kernel is(sin x)/x, and the sinc function kernel (sin x)/x is of great mathematical significance. In the paper, we also extend the similar sinc function kernel (sin x)/x to the case m and s both are not zero, which interestingly turn out as some kind of Hankel transformation.

Abstract:
Associated with each linear homogeneous differential equation y(n)= ￠ ‘i=0n ￠ ’1ai(x)y(i) of order n on the real line, there is an equivalent integral equation f(x)=f(x0)+ ￠ x0xh(u)du+ ￠ x0x[ ￠ x0uGn ￠ ’1(u,v)a0(v)f(v)dv]du which is satisfied by each solution f(x) of the differential equation.

Abstract:
Firstly, we construct kernels of integral relations among solutions of the confluent Heun equation (CHE) and its limit, the reduced CHE (RCHE). In both cases we generate additional kernels by systematically applying substitutions of variables. Secondly, we establish integral relations between known solutions of the CHE that are power series and solutions that are series of special functions; and similarly for solutions of the RCHE. Thirdly, by using one of the integral relations as an integral transformation we obtain a new series solution of the spheroidal wave equation. From this solution we construct new solutions of the general CHE, and show that these are suitable for solving the radial part of the two-center problem in quantum mechanics. Finally, by applying a limiting process to kernels for the CHEs we obtain kernels for {two} double-confluent Heun equations. As a result, we deal with kernels of four equations of the Heun family, each equation presenting a distinct structure of singularities. In addition, we find that the known kernels for the Mathieu equation are special instances of kernels of the RCHE.

Abstract:
It is demonstrated that a certain integral equation can be solved using the Painleve equation of third kind. Inversely, a special solution of this Painleve equation can be expressed as the ratio of two infinite series of spheroidal functions with known coefficients.

Abstract:
We will study the entire positive solution of the geometrically and analytically interesting integral equation: with in . We will show that only when , there are positive entire solutions which are given by the closed form up to dilation and translation. The paper consists of two parts. The first part is devoted to showing that must be equal to 11 if there exists a positive entire solution to the integral equation. The tool to reach this conclusion is the well-known Pohozev identity. The amazing cancelation occurred in Pohozev’s identity helps us to conclude the claim. It is this exponent which makes the moving sphere method work. In the second part, as normal, we adopt the moving sphere method based on the integral form to solve the integral equation. 1. Introduction In this paper, we will study a very special type of the integral equation. From an analytical point of view, such equation is interesting to be studied. We should point out that even for radius case, the analysis of the equation has been already difficult. The method of moving planes has become a very powerful tool in the study of nonlinear elliptic equations; see Alexandrov [1], Serrin [2], Gidas et al. [3], and others. The moving plane method can be applied to prove the radial symmetry of solutions, and then one only needs to classify radial solutions. The method of moving planes in the integral form has been developed by Chen and Li [4]. This technique requires not only to prove that the solutions are radially symmetric, but also to take care of a possible singularity at origin. We note that in [5], Li and Zhang have given different proofs to some previous established Liouville type theorems based on the method of moving sphere. It is this method that we applied to the integral equation to capture the solutions directly. As usual, in order for such a method to work, Pohozev’s identity is a must. In our argument, we derive and use this powerful identity to conclude that the negative exponent . Then it is the method of moving spheres in the integral form which helps us to deduce the exact solution to the integral equation in this paper, instead of only getting radius solutions or proving radial symmetry of solutions. The organization of the paper is as follows. In Section 2, we prove that . In this part, we use integration by parts to derive Pohozev’s identity in the form . It is easy to observe that if solves our integral equation, then also solves the differential equation . Thus we can put the integral form into Pohozev’s identity to calculate every term. It turns out that the boundary

Abstract:
Analytical solutions to the wave equation in spheroidal coordinates in the short wavelength limit are considered. The asymptotic solutions for the radial function are significantly simplified, allowing scalar spheroidal wave functions to be defined in a form which is directly reminiscent of the Laguerre-Gaussian solutions to the paraxial wave equation in optics. Expressions for the Cartesian derivatives of the scalar spheroidal wave functions are derived, leading to a new set of vector solutions to Maxwell's equations. The results are an ideal starting point for calculations of corrections to the paraxial approximation.

Abstract:
A new development of the ``monodromy transform'' method for analysis of hyperbolic as well as elliptic integrable reductions of Einstein equations is presented. Compatibility conditions for some alternative representations of the fundamental solutions of associated linear systems with spectral parameter in terms of a pair of dressing (``scattering'') matrices give rise to a new set of linear (quasi-Fredholm) integral equations equivalent to the symmetry reduced Einstein equations. Unlike previously derived singular integral equations constructed with the use of conserved (nonevolving) monodromy data on the spectral plane for the fundamental solutions of associated linear systems, the scalar kernels of the new equations include another kind of functional parameters -- the evolving (``dynamical'') monodromy data for the scattering matrices. For hyperbolic reductions, in the context of characteristic initial value problem these data are determined completely by the characteristic initial data for the fields. In terms of solutions of the new integral equations the field components are expressed in quadratures.

Abstract:
Bethe-Salpeter equation for the massive particles with spin 1 is considered. The scattering amplitude decomposition of the particles with spin 1 by relativistic tensors is derived. The transformation coefficients from helicity amplitudes to invariant functions is found. The integral equations system for invariant functions is obtained and partial decomposition of this system is performed. Equivalent system of the integral equation for the partial helicity amplitudes is presented.

Abstract:
The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study whether the spheroidal equations have the shape-invariance property. Expanding the super-potential term by term in the parameter alpha and solving it, we find that the superpotential loses its shape-invariance property upon to the second term. This first means that we could not solve the spheroidal problems by the SUSQM; further it is not unreasonable to say they are non-solvable in some sense.

Abstract:
The monodromy transform and corresponding integral equation method described here give rise to a general systematic approach for solving integrable reductions of field equations for gravity coupled bosonic dynamics in string gravity and supergravity in four and higher dimensions. For different types of fields in space-times of $D\ge 4$ dimensions with $d=D-2$ commuting isometries -- stationary fields with spatial symmetries, interacting waves or partially inhomogeneous cosmological models, the string gravity equations govern the dynamics of interacting gravitational, dilaton, antisymmetric tensor and any number $n\ge 0$ of Abelian vector gauge fields (all depending only on two coordinates). The equivalent spectral problem constructed earlier allows to parameterize the infinite-dimensional space of local solutions of these equations by two pairs of \cal{arbitrary} coordinate-independent holomorphic $d\times d$- and $d\times n$- matrix functions ${\mathbf{u}_\pm(w), \mathbf{v}_\pm(w)}$ of a spectral parameter $w$ which constitute a complete set of monodromy data for normalized fundamental solution of this spectral problem. The "direct" and "inverse" problems of such monodromy transform --- calculating the monodromy data for any local solution and constructing the field configurations for any chosen monodromy data always admit unique solutions. We construct the linear singular integral equations which solve the inverse problem. For any \emph{rational} and \emph{analytically matched} (i.e. $\mathbf{u}_+(w)\equiv\mathbf{u}_-(w)$ and $\mathbf{v}_+(w)\equiv\mathbf{v}_-(w)$) monodromy data the solution for string gravity equations can be found explicitly. Simple reductions of the space of monodromy data leads to the similar constructions for solving of other integrable symmetry reduced gravity models, e.g. 5D minimal supergravity or vacuum gravity in $D\ge 4$ dimensions.