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On modified Green functions in exterior problems for the Helmholtz equation

The question of non-uniqueness in boundary integral equation formulations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of coefficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.