One Special Identity between the complete elliptic integrals of the first and the third kind

Abstract

I prove an identity between the first kind and the third kind complete elliptic integrals with the following form: $$\Pi({(1+x) (1-3 x)\over (1-x) (1+3 x)}, {(1+x)^3(1-3 x)\over (1-x)^3 (1+3x)})- {1+ 3 x \over 6 x} K ({(1+x)^3(1-3x)\over (1-x)^3 (1+3x)}) = 0, (0< x < 1); =-{\pi\over 12} {(x-1)^{3/2}\sqrt{1+3 x}\over x} (x<0 or x>1).$$ This relation can be applied to eliminate the complete elliptic integral of the third kind from the analytic solutions of the imaginary part of two-loop sunset diagrams in the equal mass case. The validity of this relation in the complex domain is also briefly discussed.