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Introduction and motivation: theta functions in one variable

Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

The central character in our story is the analytic function ϑ(z, τ) in 2 variables defined by
$$ \vartheta (z,\tau ) = \sum\limits_{n\varepsilon \mathbb{Z}} {exp (\pi i n^2 \tau + 2 \pi inz)} $$
where z∈ℂ and τ ɛ H, the upper half plane Im τ > 0. It is immediate that the series converges absolutely and uniformly on compact sets; in fact, if
$$ \left| {\operatorname{Im} z} \right| < c and Im \tau > \varepsilon $$
then
$$ \left| {\exp (\pi in^2 \tau + 2\pi inz)} \right| < (\exp - \pi \varepsilon )^{n^2 } .(\exp 2\pi c)^n $$
hence, if no is chosen so that
$$ (\exp - \pi \varepsilon )^{n_o } . (\exp 2\pi c) < 1, $$
then the inequality
$$ \left| {\exp (\pi in^2 \tau + 2\pi inz)} \right| < (\exp - \pi \varepsilon )^{n(n - n_o )} $$
shows that the series converges and that too very rapidly.

Keywords

Functional Equation Normal Subgroup Entire Function Meromorphic Function Modular Form 
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.

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References and Questions

  1. R. Bellman, A Brief Introduction to Theta Functions, Holt, 1961.Google Scholar
  2. A. Hurwitz, R. Courant, Vorlerungen über Allgemeine Funtionentheorie und Elliptische Funktionen, Part II, Springer-Verlag (1929).Google Scholar
  3. B. Schoeneberg, Elliptic Modular Functions: An Introduction, Springer-Verlag (Grundlehren Band 203) (1974).Google Scholar
  4. S. Lang, Elliptic Functions, Addison-Wesley (1973)Google Scholar
  5. A. Robert, Elliptic Curves, Springer-Verlag Lecture Notes 326 (1973).Google Scholar

Copyright information

© Birkhäuser Boston 2007

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