Essential Real Analysis by Michael Field

Author:Michael Field
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

6. Topics from Classical Analysis: The Gamma-Function and the Euler–Maclaurin Formula

Michael Field1, 2

(1)Engineering Mathematics Department, Merchant Venturers School of Engineering, Bristol University, Bristol, UK

(2)Department of Mathematics, Rice University, Houston, Texas, USA

In this chapter we look at two topics from classical analysis: the Gamma-function and the Euler–Maclaurin formula. Our investigation of the Gamma-function will require many of the ideas we have developed on convergence and involves infinite products, improper integrals and other techniques and results from analysis such as differentiation under the integral sign and multiple integrals. We also need some standard results on multiple integrals (in our situation these results are elementary as we almost always assume rectangular domains and continuous, even smooth, integrands—see Exercises 2.8.10(10)). The Euler–Maclaurin formula is easy to prove but has powerful applications to estimation and asymptotics. For example, using the Euler–Maclaurin formula, we prove Stirling’s formula (estimating n! ) and also estimate Euler’s constant γ and the sums of various infinite series.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.