第一类与第三类完全椭圆积分的一个特殊恒等式
One Special Identity between the complete elliptic integrals of the first and the third kind
摘要：
证明了第一类和第三类完全椭圆积分之间的一个恒等式：$$Π({(1+x) (1-3 x)\over (1-x) (1+3 x)}, {(1+x)^3(1-3 x)\over (1-x)^3 (1+3x)})- {1+ 3 x \over 6 x} K ({(1+x)^3(1-3x)\over (1-x)^3 (1+3x)}) = 0, (0< x < 1); =-{π\over 12} {(x-1)^{3/2}\sqrt{1+3 x}\over x} (x<0 or x>1).$$这个关系式可用于从等质量双环日落图虚部的解析解中消去第三类完全椭圆积分。文中还简要讨论了这种关系在复域中的有效性。
英文摘要：
I prove an identity between the first kind and the third kind complete elliptic integrals with the following form: $$Π({(1+x) (1-3 x)\over (1-x) (1+3 x)}, {(1+x)^3(1-3 x)\over (1-x)^3 (1+3x)})- {1+ 3 x \over 6 x} K ({(1+x)^3(1-3x)\over (1-x)^3 (1+3x)}) = 0, (0< x < 1); =-{π\over 12} {(x-1)^{3/2}\sqrt{1+3 x}\over x} (x<0 or x>1).$$ 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.
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