Study of Complete Elliptic Integrals with a goal of Improving Computation of Axisymmetric Fields

In recent years I have been investigating the Complete Elliptic Integral of the First Kind, K, with a goal of finding a rapidly convergent power series R such that K can be written in terms R with the non-analyticities subtracted off or divided out. The singularities in K are logarithmic in nature, so that power series expressions of K converge slowly. Progress in this study will be presented.