Legendre-type relations for generalized complete elliptic integrals
Shingo Takeuchi
TL;DR
This paper generalizes Legendre's relation for elliptic integrals using generalized trigonometric functions, providing an alternative proof to Elliott's identity and expanding the theoretical understanding of elliptic integrals.
Contribution
It introduces a generalized form of Legendre's relation for complete elliptic integrals and offers a novel proof method based on generalized trigonometric functions.
Findings
Generalized Legendre-type relation established
Alternative proof to Elliott's identity provided
Enhanced theoretical framework for elliptic integrals
Abstract
Legendre's relation for the complete elliptic integrals of the first and second kinds is generalized. The proof depends on an application of the generalized trigonometric functions and is alternative to the proof for Elliott's identity.
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