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independent variables example
The Problem of Cauchy for Hyperbolic Equations
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adjoint analytic applications approximations arbitrary assume Bessel's inequality boundary conditions boundary value problem bounded canonical form Cauchy conditions Cauchy problem characteristic functions characteristic values circle coefficients consider constant continuous functions contour curve defined difference equation differential equation Dirichlet problem discontinuity domain dxdy example exists expression finite number follows Fourier series Fourier transform given gives Green's formula Green's function harmonic function Hence holds initial conditions initial values interval Laplace transform limit method notation obtain operator orthogonal partial derivatives plane potential proof prove radius random walk reduced region replace respect result right member second order self-adjoint sequence Similarly solution of 3.10 solve sphere string Substituting sufficiently small suppose tangential derivative theorem three dimensions trigonometric series uniformly convergent unique solution uniquely determined values and functions vanish variables velocity vibrations wave equation whence x-axis zero