In practice it is generally as inverse functions of elliptic integrals that elliptic functions are needed; and so a simple demonstration of this important property of the functions is to be desired. The important step in such a demonstration is a proof that the inverse of an elliptic integral is a one-valued function, which for every value of the variable is regular or has at worst a pole—that is, is meromorphic over the complete plane (save possibly at ∞).