On a theory of elliptic functions based on the incomplete integral of the hypergeometric function {_{2}}F_{1}(\frac{1}{4},\frac{3}{4};\frac{1}{2};z)

Using the properties of conformal mappings and differential equations, we develop a class of elliptic functions associated with the hypergeometric function ${_{2}}F_{1}(\frac{1}{4},\frac{3}{4};1;z)$ . A detailed comparison is made with the classical Jacobi elliptic functions. Within the frame work of this theory, we provide a proof and new insight into a set of identities of Ramanujan associated with the above hypergeometric function.