Theta Function Identities And Modular Equations

First,we give simple proofs of Ramanujan's modular equations of degree threeand five.Using the additive formula of Jacobi Theta function,we obtain elementaryproofs of Ramanujan's modular equations of degree three and five.A set of modularequation of degree three is derived by using Ramanujan theta function additive formula.Second,some modular equations of infinite production are derived by using Ra-manujan theta function additive formula.Some simple proofs of Ramanujan's modularequations of infinite production are given.Third,some theta function identities are established by applying the method ofconstructing elliptic functions that have no pole.Some new series representation forJacobi theta functions and infinite production are given.Fourth,by studying some known modular equations,we give a conjecture whichis about a series similar tape modular equations.Finally,some theta functions identities with two term related to the squares rela-tions of Jacobi theta functions are derived.As applications,some relations between thesquares of the Jacobi theta functions are given.