Advertisement

The Family of the Second Kind \(\{{\mathcal Z} (\sigma | t)\}\)

  • André VorosEmail author
Chapter
  • 778 Downloads
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 8)

Abstract

From Sect. 5.2 (and [106], with different notation), we recall the definition
$$\mathcal{Z}(\sigma | t) = \sum\limits_{k = 1}^{\infty}(\tau_{k}^{2} + t^{2})^{-\sigma}, \qquad {\rm Re} \sigma > \frac{1}{2},$$
(8.1)
valid for \(t \in \Omega_{2} \mathop {\rm = }\limits^{{\rm def}} \{t \in \mathbb{C} | t \pm {\rm i}\tau_{k} \notin \pm{\rm i}\mathbb{R}_{-} (\forall k)\},\) and these shorthand names at the two points \(t \in \Omega_{2}\) of special interest:
$$\mathcal{Z}_{0}(\sigma) \mathop {\rm = }\limits^{{\rm def}} \mathcal{Z}(\sigma | 0) \equiv (2 \cos \pi\sigma)^{-1}\mathcal{L}_{0}(2\sigma),\qquad \mathcal{Z}_{\ast}(\sigma) \mathop {\rm = }\limits^{{\rm def}} \mathcal{Z}(\sigma | \frac{1}{2}),$$
(8.2)
where the identity at t = 0 repeats the confluence relation (5.8).

Keywords

Zeta Function Principal Part Central Symmetry Double Pole Meromorphic Continuation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.CEA-Saclay Institut de Physique Théorique (IPhT) Orme des MerisiersGif-sur-YvetteFrance

Personalised recommendations