Fourier series: orthonormal series of piecewise continuous functions. Gram-Schmidt orthonormalization. Bessel's inequality. Completeness and closure. The Weierstrass approximation theorems. Convergence of Fourier series. Partial Differential equations: introduction, the Cauchy problem, characteristic lines and canonical forms, the wave equation, the Sturm-Liouville problem, the heat equation, the Laplace equation, maximum and minimum principles for harmonic functions, the Poisson integral.

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