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Derivation of Equations
The Electric Circuit
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analytic applications approximations arbitrary assume Bessel's inequality boundary conditions boundary value problem bounded canonical form Cauchy conditions Cauchy problem characteristic functions characteristic values coefficients consider constant continuous functions contour curve defined difference equation differential equation Dirichlet problem discontinuity domain dxdy example exists expression f is continuous follows Fourier series Fourier transform given gives Green's formula Green's function harmonic function Hence holds homogeneous initial conditions initial values interval Laplace transform lattice limit linear method notation obtain operator orthogonal partial derivatives plane potential proof prove radius 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 whence x-axis zero