The Watson transformation revisited

The Watson transformation is essentially the idea that a series can be converted into an integral in the complex plane via an “inverse residue” calculation. This idea has a long history of being used to convert expressions for the solutions of boundary value problems (BVPs) into alternative expressions from which it is easier to extract information about the solution. However, the technique is ad hoc, since given a series there are many different integrals whose residues are equal to that series. In this paper we show that, for the BVP on which the Watson transformation was first used, the optimal expression for the solution as an integral can be obtained in an algorithmic (as opposed to ad hoc) way using the Fokas transform method.