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We study the behaviour of the normal derivative of eigenfunctions of the Helmholtz equation inside billiards with Dirichlet boundary condition. These boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest. Therefore they form a reduced representation of the quantum system, analogous to the Poincar\'e section of the classical system. For the normal derivatives we introduce an equivalentdoi:10.1088/0305-4470/35/48/306 fatcat:qkkpjncylvhqhilvf3iuw67aay