Abstract
This chapter discusses integral representations for solutions to some differential equations that arise in wave theory. Integral representations are given for solutions to some linear elliptic partial differential equations in two independent variables, several of which arise in wave theory. These provide solutions to analytic Cauchy problems for the equations in terms of the complex Riemann function. Results are given for a class of second-order equations, and for several fourth-order equations. For the Helmholtz equation, applications to special functions, analytic continuation, and an inverse scattering problem, are described in the chapter. It is well known that solutions to elliptic partial differential equations with analytic coefficients are analytic functions of the independent variables. Such solutions may thus be extended analytically into the complex domain of these variables. The method of integral operators that exploits this idea was developed extensively by Bergman.
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