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 mathematician Srinivasa Ramanujan.
The Jacobi triple product identity then takes the form Here, the expression
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:[1] The special cases of Ramanujan's theta functions given by φ(q) := f(q, q) OEIS: A000122 and ψ(q) := f(q, q3) OEIS: A010054 [2] also have the following integral representations:[1] This leads to several special case integrals for constants defined by these functions when q := e−kπ (cf.