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.
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.
theta function explicit values). In particular, we have that [1]
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.
ISBN0-521-83357-4.
Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford: Oxford University Press.
ISBN0-19-286189-1.
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.
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.
theta function explicit values). In particular, we have that [1]
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.
ISBN0-521-83357-4.
Kaku, Michio (1994). Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford: Oxford University Press.
ISBN0-19-286189-1.