Abstract
Abstract
Glasser, in 2011, introduced a remarkable integral identity of physical interest and suggested that the evaluation
∫
0
1
/
2
k
K
2
(
k
)
d
k
=
π
G
4
provides the unique analytically tractable moment of K
2 on a sub-unit interval, where K denotes the complete elliptic integral of the first kind, and where
G
=
1
1
2
−
1
3
2
+
1
5
2
−
⋯
denotes Catalan’s constant. We show how a case of Clausen’s product identity related to Ramanujan’s series for
1
π
may be applied, via an integration argument derived from our past work in fractional analysis and Fourier–Legendre theory, to show how higher moments of K
2 on the same sub-unit interval may be evaluated analytically in terms of the Γ-function. This and Glasser’s moment formula are motivated by how closely related moment formulas for powers of K arise in the study of Feynman diagrams.