An inverse function of an Abelian integral. Abelian functions have two variables and four periods, and can be defined by Θ(v, τ;q' q) = sum_(λ = - ∞)^∞ e^(2π i v(λ + q') + π i τ(λ + q')^2 + 2π i q(λ + q')) (Baker 1907, p. 21). Abelian functions are a generalization of elliptic functions, and are also called hyperelliptic functions. Any Abelian function can be expressed as a ratio of homogeneous polynomials of the Riemann theta function .
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