Introduction and motivation: theta functions in one variable

Part of the Modern Birkhäuser Classics book series (MBC)


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
$$ \left| {\operatorname{Im} z} \right| < c and Im \tau > \varepsilon $$
$$ \left| {\exp (\pi in^2 \tau + 2\pi inz)} \right| < (\exp - \pi \varepsilon )^{n^2 } .(\exp 2\pi c)^n $$
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.


Functional Equation Normal Subgroup Entire Function Meromorphic Function Modular Form 
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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

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© Birkhäuser Boston 2007

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