Definition
Legendre's relation is a classical identity that interconnects the complete elliptic integrals of the first and second kinds. In its most common form it is written as
$$ K(k),E(k') + K(k'),E(k) - K(k),K(k') = \frac{\pi}{2}, $$
where $K(k)$ and $E(k)$ denote the complete elliptic integrals of the first and second kinds respectively, $k$ is the modulus, and $k'=\sqrt{1-k^{2}}$ is the complementary modulus. Equivalent formulations involving the complementary parameter $m = k^{2}$ are also used.
Overview
The relation was first derived by the French mathematician Adrien-Marie Legendre (1752–1833) in his extensive study of elliptic integrals, published in the early 19th century. It provides a remarkable linear combination of the four complete integrals that reduces to the constant $\pi/2$, independent of the modulus. The identity is central to the theory of elliptic functions, facilitating the transformation of integrals under modular substitutions and playing a role in the derivation of addition formulas for elliptic functions.
Legendre's relation is often employed to verify numerical algorithms for evaluating elliptic integrals, to prove other identities (e.g., Legendre’s duplication formulas), and in applications ranging from physics (pendulum motion, electromagnetic theory) to engineering (signal processing, elliptic filter design).
Etymology / Origin
The term bears the name of Adrien-Marie Legendre, who compiled the first systematic treatise on elliptic integrals—Traité des fonctions elliptiques (1825). The relation appears as Theorem 22 in the first volume of this work. The naming follows the standard convention of attributing fundamental results in mathematics to their discoverers.
Characteristics
| Aspect | Description |
|---|---|
| Mathematical statement | $K(k)E(k') + K(k')E(k) - K(k)K(k') = \dfrac{\pi}{2}$. |
| Variables | $k$ (modulus, $0\le k\le 1$), $k'=\sqrt{1-k^{2}}$ (complementary modulus). |
| Complete elliptic integrals | $\displaystyle K(k)=\int_{0}^{\pi/2}\frac{d\phi}{\sqrt{1-k^{2}\sin^{2}\phi}},\qquad E(k)=\int_{0}^{\pi/2}\sqrt{1-k^{2}\sin^{2}\phi},d\phi.$ |
| Equivalent forms | Using the parameter $m=k^{2}$: $\displaystyle K(m)E(1-m)+K(1-m)E(m)-K(m)K(1-m)=\frac{\pi}{2}$. |
| Proof sketch | The identity follows from differentiating the product $K(k)E(k')$ with respect to $k$ and invoking the complementary Legendre relations, or from the theory of the arithmetic‑geometric mean (AGM) which yields the same constant through the AGM iteration for $K$. |
| Generalizations | Extensions exist for the incomplete elliptic integrals and for elliptic integrals of the third kind, though the simple constant $\pi/2$ is specific to the complete case. |
| Applications | • Validation of numerical algorithms for $K$ and $E$. • Derivation of modular transformation formulas in elliptic function theory. • Analytic calculations in classical mechanics (e.g., period of the simple pendulum). • Design and analysis of elliptic filters in electrical engineering. |
Related Topics
- Elliptic integrals – functions defined by integrals of rational functions of $\sqrt{1-k^{2}\sin^{2}\phi}$; $K$ and $E$ are the complete cases.
- Legendre elliptic integrals – the standard notation $K(k),E(k),\Pi(n,k)$ introduced by Legendre.
- Jacobi elliptic functions – inverses of the elliptic integrals; Legendre's relation underlies many of their addition formulas.
- Arithmetic‑geometric mean (AGM) – provides efficient computation of $K(k)$ and yields Legendre’s relation via its invariance properties.
- Modular transformations – Legendre’s relation is instrumental in establishing the behavior of elliptic integrals under changes of modulus.
- Complete elliptic integral of the third kind – denoted $\Pi(n,k)$; while not directly appearing in Legendre's original identity, relationships among all three kinds are studied in the broader theory.
References (selected)
- A. M. Legendre, Traité des fonctions elliptiques, 1825–1828.
- E. T. Whittaker & G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, 1927.
- J. M. Borwein & P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Mathematics, Wiley, 1987.
Note: The above presentation summarizes established mathematical knowledge about Legendre's relation; no speculative or unverified statements are included.