Approximate Solutions to Complete Elliptic Integrals for Practical Use in Water Engineering

Complete elliptic integrals have many applications in water engineering. Examples can be found in different fields such as hydrodynamics, water wave mechanics, groundwater engineering, sediment transport, irrigation and drainage engineering, and shallow water and deep water engineering. In practice, these integrals are evaluated from lookup tables or closed-form solutions (infinite series). Most tables use discrete values of the modulus, which makes accurate interpolation difficult. On the other hand, a series expansion can only be applied to a limited range of the modulus, and is not suitable for manual calculations. Consequently, it is of interest to approximate complete elliptic integrals by simple and accurate algebraic formulas over the entire practical range of the modulus. In current research, the undetermined coefficients method is used for this purpose. Various techniques can be used to determine the unknown coefficients of this method, but most of these do not depend on the integrand. Curve fitting, requiring exact numerical evaluation of the integrals followed by nonlinear optimization, is used as a powerful tool to determine the optimal values of the unknown coefficients as a function of the integrand. The proposed approximations are simple and valid over the full range $0\le k\le 0.99$ with maximum relative errors of 0.31 and 0.12% for the complete elliptic integrals of the first and second kinds, respectively. More complex approximations achieving higher accuracy with percentage errors lower than 0.07% for $0\le k\le 0.99$, are also derived. This approach can also be applied to similar integrals of constant integration limits. As such, it should be of interest to practitioners in the water engineering community.