ζ-Function method for infinite series

✍ Scribed by Alfred Actor

• Book ID: 104758914
• Publisher: Springer
• Year: 1986
• Tongue: English
• Weight: 239 KB
• Volume: 12
• Category: Article
• ISSN: 0377-9017
• DOI: 10.1007/bf00402669

✦ Synopsis

The ~-function method is used to rearrange Dirichlet series of the form E,.( + ) r a m -S g ( x / m ) into power series in x. This displays explicitly the analyticity in s of the series. Generalized ~-functions of physical interest can be analysed by this method.

Recently, a new method [ 1 ] was introduced for rearranging a large class of infinite series, including many Fourier and Bessel function series, into power-series form. This method uses in an essential way the properties of the Riemann ~-function. The exact hightemperature expansions of the Bose and Fermi thermodynamic potentials for arbitrary mass and (complex) chemical potential are readily obtained [1] by the (-function method. (See [2] for a fuller account of this problem, as well as of the general method.) In the present article we apply the same technique to a quite different class of infinite series, and obtain some rather general mathematical results. At the conclusion of the article we briefly discuss their application to ~-functions of physical interest.