In his last letter to Hardy, Ramanujan defined 17 Jacobi theta function-like functions F(q) with left bracketing bar q right bracketing bar <1 which he called "mock theta functions" (Watson 1936ab, Ramanujan 1988, pp. 127-131; Ramanujan 2000, pp. 354-355). These functions are q-series with exponential singularities such that the arguments terminate for some power t^N. In particular, if f(q) is not a Jacobi theta function, then it is a mock theta function if, for each root of unity ρ, there is an approximation of the form f(q) = sum_(μ = 1)^M t^(k_μ) exp( sum_(ν = - 1)^N c_μν t^ν) + O(1) as t->0^+ with q = ρ e^(-t).

Jacobi theta functions | Mordell integral | q-series

Leila Dragonette | Srinivasa Aiyangar Ramanujan