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SUPERPOSITION OF SOLUTIONS
ORTHOGONAL SETS OF FUNCTIONS
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axis Bessel's equation boundary conditions boundary value problem bounded coefficients constant continuous function converges uniformly cosine series cylinder defined denote domain eigenfunctions eigenvalues exist finite number follows formal solution Fourier cosine series Fourier series Fourier sine series Fourier theorem function f(x harmonic function heat equation Hence homogeneous conditions independent variables infinite initial temperature inner product insulated integrand interior interval 0,c Jn(x Laplace's equation Legendre's linear combination linear homogeneous nirx Note nx dx one-sided derivatives orthogonal set orthonormal set partial differential equations periodic extension periodic function plane Pn(x polynomials positive number positive roots represents satisfies the conditions second order sectionally continuous function separation of variables series converges series representation set of functions sinh slab solid string Sturm-Liouville problem surface temperature zero tion transverse displacements uniform convergence vector Verify write written