Singular Euler-Maclaurin expansion
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We generalise the Euler-Maclaurin expansion and make it applicable to the product of a differentiable function and an asymptotically smooth singularity. The difference between sum and integral is written as a differential operator acting on the non-singular factor only plus a remainder integral. The singularity can be included in generalised Bernoulli polynomials which form the coefficients of the differential operator and determine the integrand of the remainder integral. As the singularity is being integrated instead of being differentiated, the convergence of our expansion mainly depends on the growth rates of the derivatives of the differentiable factor. If the non-singular function is of suitably small exponential type, the expansion order can be taken to infinity, avoiding the divergence of the standard Euler-Maclaurin expansion. A closed form for the differentiable operator of infinite order, finite order approximations thereof, as well as a generating function for the generalised Bernoulli polynomials are given. We demonstrate the numerical performance of the singular Euler-Maclaurin expansion by applying it to the computation of non-linear long-range forces inside a macroscopic one-dimensional crystal with 2× 10^10 particles
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