The numerical evaluation of expressions involving complete elliptic integrals

Formulas involving the complete elliptic integrals K, E, and II occur in the calculation of standards of inductance and mutual inductance, in the calculation of the magnetic field components of coils, and in the calculation of the force between coils. Other examples are found in such fields as hydrodynamics, electrostatics, and in certain applications of conformal mapping. The elliptic integrals may appear explicitly in the formulas or the computer may have to evaluate a complicated integral expression which is recognized as elliptic in nature, but in which K, E, and II do not appear. Especial attention will here be paid to the procedure to be adopted for reducing such expressions to the standard forms.