Abstract
The main object of the present paper is to derive a unification (and generalization) of certain interesting formulae from Barton (1983 Proc. R. Sec. A 388 445-56) and Bushell (1987 Math. Proc. Cambridge Phil. Sec. 101 1-5) associated with the complete elliptic integrals of the first and second kind. Relevant connections between the results presented here with those given earlier by other workers on the subject are also pointed out. Various families of generalized elliptic integrals, and indeed also definite integrals of such families with respect to their modulus, are known to arise (among other places) in the studies of crystallographic minimal surfaces (cf, e.g., Cvijovic and Klinowski (1994 Proc. R. Sec. A 444 525-32)) and in the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk (cf Bjorkberg and Kristensson (1987 Can. J. Phys. 65 723-34)).