Charge density estimations for particle beams based on orthogonal polynomial series
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A beam's charge density, treated as a smooth and continuous function, can be approximated using polynomial orthogonal series. This will allow a solution of Poisson's equation to be found. Obtaining the most accurate solution to the space-charge potential requires the best approximated charge density. Several beam distributions are approximated using the classic Jacobi polynomials generated by the traditional recursion relations and the moment method. Varying the particle number and the order of the approximation allows to compare the performance of the different polynomials and to determine if a particular combination of both works best. Although the three polynomials studied give similar results, with Legendre being slightly better, the approximation coefficients should be allowed to converge and taken to high orders for best results.