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: $$\Pi (\frac{(1+x)(1-3x)}{(1-x)(1+3x)},\frac{(1+x{)}^3(1-3x)}{(1-x{)}^3(1+3x)})-\frac{1+3x}{6x}K(\frac{(1+x{)}^3(1-3x)}{(1-x{)}^3(1+3x)})=0,(0<x<1);=-\frac{\pi }{12}\frac{(x-1{)}^{3/2}\sqrt{1+3x}}{x}(x<0orx>1).$$ 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.

Comments: 8 pages, 2 figures