Abstract
This chapter is devoted to two topics from classical (pre-twentieth century) analysis: the Gamma-function and the Euler–Maclaurin formula. The analysis of the Gamma-function draws on many results from the preceding chapters and requires new tools such as differentiation under the integral sign and convexity. After defining Bernoulli numbers and polynomials and computing their Fourier series, a proof is given of the Euler–Maclaurin formula and a number of applications are given including Stirling’s formula, estimates of Euler’s constant and the sums of infinite series (such as the infinite sum of n −2).
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Field, M. (2017). Topics from Classical Analysis: The Gamma-Function and the Euler–Maclaurin Formula. In: Essential Real Analysis. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-67546-6_6
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