The general analytic solution to the functional equation (equation presented) is characterized. Up to the action of the symmetry group, this is described in terms of Weierstrass elliptic functions. We illustrate our theory by applying it to the classical addition theorems of the Jacobi elliptic functions and the functional equations φ1 (x + y) = φ4(x)φ5(y) + φ4(y)φ5(x) and ψ1(x + y) = ψ2(x + y)φ2(x)φ3(y) + ψ3(x + y)φ4(x)φ5(y).
- Functional equation
- Special functions