Theta-function identities and the explicit formulas for theta-function and their applications

We define two quotients of theta-functions depending on two positive real parameters. We then show how they are connected with two parameters of Dedekind eta-function and the Ramanujan– Weber class invariants. Explicit formulas for determining values of the theta-function is derived, and several examples will be given and using them, we give some complete explicit results for the complete elliptic integral of the first kind and the Gaussian hypergeometric function. Also several new modular equations for the theta-function are derived.  2004 Elsevier Inc. All rights reserved.