Theta Identities And Their Applications

In this paper we prove a generalized Ramanujan's circular summation of theta function using the theory of elliptic and theta functions From this identity we can get many theta identities including an addition formula of theta functions.Besides,we can get some new trigonometric identities.Starting from two theta identities respectively,we reobtain some results of representing integers as sums of squares and some conclusions of modular equation.