Remarks on Evaluating Elliptic Integrals of Gradshteyn and Ryzhik

Most of the elliptic integrals in Gradshteyn and Ryzhik are in sections 3.13-3.18, but there are many others scattered throughout the book. GR groups them, as they do most integrals, by the appearance of the integrand. This is unfortunate, as the substitutions required to evaluate these integrals do not conform to this scheme. More on this below.

All books that cover the evaluation of elliptic integrals provide some guidance regarding the required substitution. Some books provide a generic form with unknown parameters, while others provide specific substitutions for specific type of integrals.

In Handbook of Elliptic Integrals for Engineers and Scientists, the authors, Byrd and Friedman, use the following approach to evaluate elliptic integrals. First, they provide values for the unknown parameters of generic substitutions mentioned above. Next, they group integrals into sections based on a single substitution. These integrals are then reduced to integrals of functions of Jacobi elliptic functions. Such integrals appear repeatedly throughout the book, thus BF then provide evaluations of these integrals of functions of Jacobi elliptic functions in other sections. However, they do not provide the final answer in terms of the original variables (GR does do this).

On this blog I have been establishing results that will lead to evaluation of elliptic integrals. I have been working through the BF procedure in reverse. Now, I will use the following steps to evaluate the elliptic integrals of GR.

I will follow the groupings used in BF and map the results to the numbering scheme of GR. Using the substitutions provided by BF, I will work out all of the substitutions required, including the derivative to change variables. This step is straightforward, but care is required to avoid using the wrong Jacobi elliptic function expression so as to achieve the desired simplification as opposed to a complicated algebraic expression of Jacobi elliptic functions. Here, I will also write out the simplified form of the relevant Jacobi elliptic functions in terms of the original variables. Again, care must be exercised to achieve desired simplified results. Now, evaluation of elliptic integrals has been reduced to using these substitutions and simplifications, combined with previous work done to derive relationships between, derivatives of, and integrals of Jacobi elliptic functions.