An approximation for zero - balanced Appell function F 1 near ( 1 , 1 )

We suggest an approximation for the zero-balanced Appell hyper-geometric function F 1 near the singular point (1, 1). Our approximation can be viewed as a generalization of Ramanujan's approximation for zero-balanced 2 F 1 and is expressed in terms of 3 F 2. We find an error bound and prove some basic properties of the suggested approximation which reproduce the similar properties of the Appell function. Our approximation reduces to the approximation of Carlson-Gustafson when the Appell function reduces to the first incomplete elliptic integral.