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analytical continuation approximation asymptotic expansion auxiliary vector space Bessel's inequality boundary conditions boundary value problem Cauchy data circle coefficients complex potential consider constant Corollary corresponding cylinder defined denote derivatives Diaz differential equation Dirichlet integral Dirichlet problem distance earth elastic electrons elliptic equality holds equals the vector field fluid formula gases given gradient Green's harmonic Hence hodograph incompressible flow intersection ionized gas ions kinetic theory known vectors Laplace's equation lifting surface lower bounds magnetic manifold Math mathematical maximum method minimum principles molecular molecules motion negative Neumann's problem non-constant function obtained orthonormal vectors particles plate poles positive integers present paper protons quadratic integral radius real numbers region satisfying scalar product Schwarz singular solution sphere streamlines subsonic supersonic temperature theorem thermal diffusion upper and lower vanishes vector of zero vector space velocity potential viscosity zero length zero vector