*(English)*Zbl 1118.33010

The authors obtain two convergent series expansions for the incomplete elliptic integral of the first kind

valid at any point in the unit square $0<\lambda ,k<1$. These expansions are expressed in terms of recursively computed elementary functions. The expansions are truncated after $N$ terms and, by expressing the tails as integrals combined with use of bounds for certain hypergeometric functions, explicit bounds for the remainders ${R}_{N}$ are obtained.

The truncated expansions yield asymptotic approximations for $F(\lambda ,k)$ as $\lambda $ and/or $k$ approach unity. The approximations also remain valid as the logarithmic singularity $\lambda =k=1$ is approached in any direction. The first two approximations complete with error bounds are presented explicitly and numerical calculations are given to illustrate their accuracy.

##### MSC:

33E05 | Elliptic functions and integrals |

33C75 | Elliptic integrals as hypergeometric functions |

33F05 | Numerical approximation and evaluation of special functions |