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The Fourier series is a method of solving partial differential equations. It finds an explicit, finite expression for the Green's function, which represents the general solution of such problems. These solutions can be used to compute solutions to many problems, such as finding eigenvalues and eigenvectors of matrices and determining low-pass filter coefficients. It is named after Jean-Baptiste Joseph Fourier who first described it in 1824. The time dependence of the solution (the Laplace transform) was first studied by Robert Hooke who gave an orderly expression for doing this about 350 years later. The original derivations were done without knowledge of group theory and modern work has seen these results generalized using group theory. Suppose that a variational problem has a solution "s" to a partial differential equation. Next, suppose that the variational problem is defined to be "P"("s", "t") = "f"("s", "t") + λ · ("s", "t") + u, where u represents a source term and λ a source-free term.If the solution is eigenfunction, then after the natural logarithm the expression becomes:where formula_3 denotes the characteristic function of all real values of λ and formula_4 denotes its imaginary part. The overall time dependence of the solution is given by the Laplace transform of the convolutionThis can be rearranged to givewhich can then be integrated over "t" to obtain the inverse Laplace transform of the overall time-dependent solution:The Fourier transform is a mapping from complex space to real space. If the series on the right has no terms with negative real part, i.e. divided by formula_6, then its inverse Fourier transform is given by where "h" = 1/2 on (0, ±1). The general solution is given by the inverse Laplace transform:An advantage of using the Fourier series method is that we can calculate it very accurately. There are three methods: the power series, the Wronskian and elementary methods. Which one we use depends on how much accuracy we need and how difficult it is to calculate either P("s", "t") or "s"("t") for our problem. For example; if "P"("s", "t") = P("s"), then we can easily find u by taking the difference between both sides of the equation and equating like terms. The same idea can be used to find "s". If "s"("t") = P("s") then we get "s" = -P("s").The series depends on the boundary conditions. To simplify this we can use normal modes where we assume that all dependent variables in our problem are zero when the independent variable is zero, and we assume that all dependent variables in our problem are zero when the independent variable is equal to infinity:Notice that the series converges faster in the boundary conditions and faster in x and y than when values at infinity are included. We can also use periodic boundary conditions; these depend on our application.

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