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 
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/
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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.

[1] I. N. Vekua, “O metagarmonicheskikh funktsiyakh”, Trudy Tbilisskogo matem. in-ta, 12 (1943), 105–166

[2] L. A. Muravei, “Analiticheskoe prodolzhenie po parametru funktsii Grina vneshnikh kraevykh zadach dlya dvumernogo uravneniya Gelmgoltsa, I”, Matem. sb., 97(139) (1975), 403–433 | MR

[3] L. A. Muravei, “Asimptoticheskoe povedenie pri bolshikh znacheniyakh vremeni reshenii vtoroi i tretei vneshnikh kraevykh zadach dlya volnovogo uravneniya s dvumya prostranstvennymi peremennymi”, Trudy Matem. in-ta im. V. A. Steklova, CXXVI (1973), 73–144 | MR

[4] F. W. Olver, “The asymptotic solution of linear differential equations of the second order for large values of a parameter”, Philos. Trans. Roy. Soc., London, 247 (1954), 307–369 | DOI | MR

[5] G. N. Vatson, Teoriya besselevykh funktsii, IL, Moskva, 1949

[6] I. S. Grandshtein, I. M. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, Moskva, 1962