Multi-parameter analysis of the obstacle scattering problem

We consider the acoustic field scattered by a bounded impenetrable obstacle and we study its dependence upon certain set of parameters. As usual, problem is modeled an exterior Dirichlet for Helmholtz equation $\Delta u +k^2u=0$. show that solution $u$ far pattern $u_\infty$ depend real analytically on shape obstacle, wave number $k$, datum. also prove similar result corresponding Dirichlet-to-Neumann map.