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
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
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:
We have the following integral representation for the full two-parameter form of Ramanujan's theta function:
The special cases of Ramanujan's theta functions given by φ(q) := f(q, q)
A000122 and ψ(q) := f(q, q3)
 also have the following integral representations:
This leads to several special case integrals for constants defined by these functions when q := e−kπ (cf.
theta function explicit values). In particular, we have that
Application in string theory
The Ramanujan theta function is used to determine the
critical dimensions in
Bosonic string theory,
superstring theory and
- Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 32. Cambridge: Cambridge University Press.
- Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. Vol. 96 (2nd ed.). Cambridge: Cambridge University Press.
Encyclopedia of Mathematics,
EMS Press, 2001 
- Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford: Oxford University Press.
Weisstein, Eric W.
"Ramanujan Theta Functions".