Theta function: Difference between revisions

Latest revision as of 06:01, 28 October 2024

Citable Version ^[?]

This editable Main Article is under development and subject to a disclaimer.

[edit intro]

In mathematics, a theta function is an analytic function which satisfies a particular kind of functional equation. Theta functions occur as generating functions associated to quadratic forms. They are modular forms of half-integer weight.

Jacobi's theta function is a power series in $\exp(\pi i\tau )$ (traditionally referred to as the nome)

\vartheta (\tau )=\sum _{n\in \mathbf {Z} }\mathrm {e} ^{\pi in^{2}\tau }.\,

Jacobi's triple product identity

\prod _{m=1}^{\infty }(1-x^{2n})(1+x^{2n-1}z^{2})(1+x^{2n-1}z^{-2})=\sum _{m\in \mathbf {Z} }x^{m^{2}}z^{2m}.

Tom M. Apostol (1976). Introduction to Analytic Number Theory. Springer-Verlag. ISBN 0-387-90163-9.
Tom M. Apostol (1990). Modular functions and Dirichlet Series in Number Theory, 2nd ed. Springer-Verlag. ISBN 0-387-97127-0.

@@ Line 1: / Line 1: @@
-In [[mathematics]], a '''theta function''' is an analytic function which satisfies a particular kind of [[functional equation]].
+{{subpages}}
+In [[mathematics]], a '''theta function''' is an analytic function which satisfies a particular kind of [[functional equation]].  Theta functions occur as [[generating function]]s associated to [[quadratic form]]s.  They are [[modular form]]s of half-integer [[weight of a modular form|weight]].
+Jacobi's theta function is a [[power series]] in <math>\exp(\pi i \tau)</math> (traditionally referred to as the ''nome'')
+:<math>\vartheta(\tau) = \sum_{n \in \mathbf{Z}} \mathrm{e}^{\pi i n^2 \tau} .\,</math>
+Jacobi's triple product identity
+:<math>\prod_{m=1}^\infty(1-x^{2n})(1+x^{2n-1}z^2)(1+x^{2n-1}z^{-2}) = \sum_{m \in \mathbf{Z}} x^{m^2}z^{2m} .</math>
+==References==
+* {{cite book | author=Tom M. Apostol | title=Introduction to Analytic Number Theory | series=Undergraduate Texts in Mathematics | year=1976 | publisher=[[Springer-Verlag]] | isbn=0-387-90163-9 }}
+* {{cite book | author=Tom M. Apostol | title=Modular functions and Dirichlet Series in Number Theory  | edition=2nd ed | series=[[Graduate Texts in Mathematics]] | volume=41 | publisher=[[Springer-Verlag]] | year=1990 | isbn=0-387-97127-0 }}
+[[Category:Suggestion Bot Tag]]