Euler‐Maclaurin Summation of Trigonometric Fourier Series

Trigonometric Fourier series are, in general, difficult to sum to high accuracy. An example is given by the series in which α and β (>0) are rational numbers satisfying 0< β / α ≤1, where λ is an independent variable and j is a positive integer or zero. This paper presents a method for the efficient evaluation of the sum of such series. Fourier series which are the real or the imaginary part of , but which are not explicitly expressible as simple polynomials in λ , are obtained as the sum of a logarithic term and an infinite series in powers of λ , whose expansion is valid when 0< λ ≤(2π/ α ) and is exact . When the Fourier series is expressible as a polynomial in λ , the method identifies that polynomial.