The Riemann theta function is a complex function of g complex variables that occurs in the construction of quasi-periodic solutions of various equations in mathematical physics . Any Abelian function can be expressed as a ratio of homogeneous polynomials of the Riemann theta function . Let the imaginary part of a g×g matrix F be positive definite, and m = (m_1, ..., m_g) be a row vector with coefficients in Z. Then the Riemann theta function is defined by ϑ(u|F) = sum_m exp[2π i(m^T u + 1/2 m F^T m)].

Abelian function | Jacobi theta functions | Ramanujan theta functions | Siegel theta function