Euler-Maclaurin formula

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Euler-Maclaurin formula

[¦ȯi·lər mə′klȯr·ən ‚fȯr·myə·lə]
(mathematics)
A formula used in the numerical evaluation of integrals, which states that the value of an integral is equal to the sum of the value given by the trapezoidal rule and a series of terms involving the odd-numbered derivatives of the function at the end points of the interval over which the integral is evaluated.
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Here Varadarajan (mathematics, UCLA) takes a unique approach, combining the history of Euler's work with practice of it, covering Euler's biography, his theories, and other "teachable topics" as zeta values, Euler-Maclaurin Sum Formula, divergent series and integrals, and Euler products.
They cover the elementary methods, Bernoulli numbers, including the Riemann Zeta function and the Euler-MacLaurin sum formula, modular forms and Hecke's theory of modular forms, representations of numbers as sums of squares, including the singular series and Liouville's methods and elliptical modular forms, arithmetic progression, and applications such as computing sums of two to four squares, resonant cavities and diamond cutting.