Theta function

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

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

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.