On an eigenfunction expansion associated with a condition of radiation

Abstract

In this paper certain Bessel type eigenfunction expansions are developed by considering a non-seif-adjoint problem which involves a radiation type condition

On an integral transform associated with a condition of radiation, part 2

“…All solutions of Bessel's equation 4are 0(r~i) as r -> oo so that riy = 0 (1). Therefore, by (3), the above expression reduces to…”

“…This expansion was discussed in (3) where it was pointed out that the above series is not usually convergent. If the series is not convergent, an alternative representation of f(r) exists but it is not an expansion in terms of the eigenfunctions H'^(kr) and in general it cannot be obtained from the values of F-y(u n ).…”

On an inversion formula

“…This transform appeared in connection with an earlier investigation [4] in which an attempt was made to devise an integral transform that could be adapted to the solution of certain boundary value problems involving the space form of the wave equation and the condition of radiation:…”

“…The integral formula derived in [4] is given by the equation where L denotes the imaginary axis of the complex u -plane and C denotes a loop enclosing the positive real axis. The object in constructing the formula (4) was to generate the type of expansion which appears in the theory of diffraction and which involves the eigenfunctions H ( J?…”

On an integral transform

“…This method is described in [4]. Instead the method developed by the author to invert the transform (3) has been applied.…”