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Phase Transitions and Renormalization Group$
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Jean Zinn-Justin

Print publication date: 2007

Print ISBN-13: 9780199227198

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199227198.001.0001

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Gaussian expectation values. Steepest descent method

Gaussian expectation values. Steepest descent method

(p.19) 2 Gaussian expectation values. Steepest descent method
Phase Transitions and Renormalization Group

Jean Zinn-Justin

Oxford University Press

This chapter presents a number of technical results concerning generating functions, Gaussian measures, and the steepest descent method. It begins by introducing the notion of generating function of the moments of a probability distribution. It then calculates Gaussian integrals and prove Wick′s theorem for Gaussian expectation values, a result that is simple but of major practical importance. The steepest descent method provides asymptotic evaluations, in some limits, of real or complex integrals. It leads to calculations of Gaussian expectation values, which explains its presence in this chapter. Moreover, the steepest descent method will be directly useful in this work. Exercises are provided at the end of the chapter.

Keywords:   generating functions, Gaussian expectation values, connected contributions, Wick′s theorem, Gaussian integrals

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