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A Simple Proof of an Inequality for the Complete Elliptic Integral of the First Kind

29 April 2025, Version 1
Working Paper
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

This article presents a simple proof of inequalities for the complete elliptic integral of the first kind through symmetry arguments and interval reduction techniques. Furthermore, it provides a geometric interpretation of the bounds for this integral, offering intuitive insights into its analytic behavior.

Keywords

Complete elliptic integral
First kind elliptic integral
Symmetry arguments
Inequalities

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Version History

Apr 29, 2025 Version 1

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The content is available under CC BY NC ND 4.0 [opens in a new tab]

DOI

10.33774/coe-2025-dq13h
D O I: 10.33774/coe-2025-dq13h [opens in a new tab]

Author’s competing interest statement

The author(s) have declared they have no conflict of interest with regard to this content

Ethics

The author(s) have declared ethics committee/IRB approval is not relevant to this content

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