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Adjusted Forms of the Fourier Coefficient Asymptotic Expansion and Applications in Numerical Quadrature
J. N. Lyness
Mathematics of Computation
Vol. 25, No. 113 (Jan., 1971), pp. 87-104
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2005134
Page Count: 18
You can always find the topics here!Topics: Fourier coefficients, Mathematical integrals, Analytics, Numerical quadratures, Sine function, Mathematical functions, Coefficients, Rectangles, Asymptotic value, Algebra
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The conventional Fourier coefficient asymptotic expansion is derived by means of a specific contour integration. An adjusted expansion is obtained by deforming this contour. A corresponding adjustment to the Euler-Maclaurin expansion exists. The effect of this adjustment in the error functional for a general quadrature rule is investigated. It is the same as the effect of subtracting out a pair of complex poles from the integrand, using an unconventional subtraction function. In certain applications, the use of this subtraction function is of practical value. An incidental result is a direct proof of Erdélyi's formula for the Fourier coefficient asymptotic expansion, valid when f(x) has algebraic or logarithmic singularities, but is otherwise analytic.
Mathematics of Computation © 1971 American Mathematical Society