Ramanujan theta function
In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after Srinivasa Ramanujan.
Definition
The Ramanujan theta function is defined as
for |ab| < 1. The Jacobi triple product identity then takes the form
Here, the expression denotes the q-Pochhammer symbol. Identities that follow from this include
and
and
this last being the Euler function, which is closely related to the Dedekind eta function. The Jacobi theta function may be written in terms of the Ramanujan theta function as:
Application in string theory
The Ramanujan theta function is used to determine the critical dimensions in Bosonic string theory, superstring theory and M-theory.
References
- Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics. 32. Cambridge: Cambridge University Press.
- Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. 96 (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-83357-4.
- Hazewinkel, Michiel, ed. (2001), "Ramanujan function", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford: Oxford University Press. ISBN 0-19-286189-1.
- Weisstein, Eric W. "Ramanujan Theta Functions". MathWorld.