Theta function: Difference between revisions
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imported>Richard Pinch (expanded somewhat, References to Apostol) |
imported>Richard Pinch (added Jacobi's triple product identity) |
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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]]. | 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 | Jacobi's theta function is a [[power series]] in <math>\exp(\pi i \tau)</math> (traditionally referred to as the ''nome'') | ||
:<math>\vartheta( | :<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== | ==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=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 }} | * {{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 }} | ||
Latest revision as of 04:13, 3 January 2009
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 (traditionally referred to as the nome)
Jacobi's triple product identity
References
- 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.