M. A. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, 10th printing, 1972.

D. Bouche and O. Lafitte, Comparaison couche limite analyse microlocale Asymptotic Analysis

H. Boumaza and O. Lafitte, The band spectrum of the periodic Airy-Schrödinger operator on the real line, J. Differential Equations, vol.264, issue.1, pp.455-505, 2018.

M. Cessenat, Mathematical methods in electromagnetism. Linear theory and applications, Series on Advances in Mathematics for Applied Sciences, vol.41, 1996.

D. Colton and P. Monk, The Scattering of Electromagnetic Waves by a Perfectly Conducting Infinite Cylinder Mathematical, Methods in the Applied Sciences, vol.12, pp.503-518, 1990.

, Digital Library of Mathematical Functions

, On the effect of the boundary conditions on the eigenvalues of ordinary differential equations Contributions to analysis and geometry, pp.139-150, 1981.

. Harrell, The band-structure of a one-dimensional, periodic system in a scaling limit, Ann. Physics, vol.119, issue.2, pp.351-369, 1979.

O. Lafitte, Diffraction in the high frequency regime by a thin layer of dielectric material I, II: the equivalent boundary condition SIAM, J. Appl. Maths, vol.59, issue.3, pp.1020-1052, 1999.

O. Lafitte, Hybrid singularity for the Oblique Incidence Response of a Cold Plasma Indiana University Math, Journal, vol.69, issue.4, pp.1317-1363, 2020.

O. Lafitte, M. Williams, and K. Zumbrun, The Erpenbeck high frequency instability theorem for Zeldovitch-von Neumann-Döring detonations, Arch. Ration. Mech. Anal, vol.204, issue.1, pp.141-187, 2012.

S. Nosal, P. Soudais, and J. Greffet, Integral Equation Modeling of Doubly Periodic Structures With an Efficient PMCHWT Formulation, IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, vol.60, issue.1, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01636345

F. J. Olver, Uniform Asymptotic Expansions of Solutions of Linear Second-Order Differential Equations for Large Values of a Parameter Philosophical Transactions of the, Mathematical and Physical Sciences, issue.984, pp.479-517, 1958.

F. J. Olver, Asymptotics and special functions, 1997.

P. Payen, , 2020.

B. Stupfel, Improved transmission conditions for a one-dimensional domain decomposition method applied to the solution of the Helmholtz equation, Journal of Computational Physics, vol.229, pp.851-874, 2010.

A. Sharples, Uniform asymptotic expansions of modified Mathieu functions, J. Reine Angew. Math, vol.247, pp.1-17, 1971.

A. Sharples, Uniform asymptotic forms of modified Mathieu functions, Quart. J. Mech. Appl. Math, vol.20, pp.365-380, 1967.