An Important Theta Function Identity

In this article,we use the theta functions with rational characteristics to construct an elliptic function,then apply the residue theorem to the elliptic functions to get a theta function identity with five variables.With using the special relations between the four variables,we have four important theta function identities.By these identities,we not only obtain some identities relating toφ(q),ψ(q) and Lambert series,but also obtain four identities associated with the modular equations of degree 5.