Compact extended algorithms for elliptic integrals in electromagnetic field and potential computations. I. Elliptic integrals of the first and second kind with extended integration range

Electromagnetic field and potential computations on elements with curved contours in boundary element and volume integral methods require evaluation of a number of Jacobian complete and/or incomplete elliptic integrals of all three kinds for the same modulus but different angles, depending on the arc length and the angle coordinate of the field point. Up to now, they have been evaluated individually, repeating the same algorithms a number of times. To reduce such redundant computations, a compact algorithm based on a modified Landen transformation is developed for elliptic integrals with an extended integration range - pi i\n\n\t\t