Geodesic
On the Green function of a slightly heterogeneous waveguide
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 13, Tome 128 (1983), pp. 139-148

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The uniform asymptotic expansion on powers of a small parameter $\varepsilon$ for a solution of the problem is $[\Delta+k^2(\varepsilon x, y)]u(x, y)=\delta(x-x_0)\delta(y-y_0)$, $u(x, 0)=u(x, H)=0$ obtained. 
@article{ZNSL_1983_128_a14,
     author = {N. A. Razumovskii},
     title = {On the {Green} function of a slightly heterogeneous waveguide},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {139--148},
     publisher = {mathdoc},
     volume = {128},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_128_a14/}
}
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N. A. Razumovskii. On the Green function of a slightly heterogeneous waveguide. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 13, Tome 128 (1983), pp. 139-148. http://geodesic.mathdoc.fr/item/ZNSL_1983_128_a14/