Geodesic
Analytic continuations with respect to a~parameter of the Green function of exterior boundary value problems for the two-dimensional Helmholtz equation.~II
Sbornik. Mathematics, Tome 30 (1976) no. 1, pp. 77-118

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This paper gives a construction of some special potentials for the two-dimensional Helmholtz equation. With these potentials, one can establish the existence of Green's functions of exterior boundary-value problems for each $k$ from the complex upper $k$-plane, and the analyticity of these functions in that region. More than that, one can establish the existence of an analytic continuation to the region $$ \{0>\operatorname{Im}k>-\beta(1+|\operatorname{Re}k|^{1/3},\,|\operatorname{Re}k|>0\} $$ for some $\beta>0$, with estimates characterizing the behavior of the Green functions for large absolute values of $k$. Bibliography: 6 titles. 
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     author = {L. A. Muravei},
     title = {Analytic continuations with respect to a~parameter of the {Green} function of exterior boundary value problems for the two-dimensional {Helmholtz} {equation.~II}},
     journal = {Sbornik. Mathematics},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1976_30_1_a5/}
}
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L. A. Muravei. Analytic continuations with respect to a~parameter of the Green function of exterior boundary value problems for the two-dimensional Helmholtz equation.~II. Sbornik. Mathematics, Tome 30 (1976) no. 1, pp. 77-118. http://geodesic.mathdoc.fr/item/SM_1976_30_1_a5/