An approximation for zero-balanced Appell function F1 near (1,1)

We suggest an approximation for the zero-balanced Appell hypergeometric function F1 near the singular point (1,1). Our approximation can be viewed as a generalization of Ramanujan's approximation for zero-balanced and is expressed in terms of . 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.