Jacobi Identities

"The" Jacobi identity is a relationship

(1)

between three elements , , and , where  is the commutator. The elements of a Lie algebra satisfy this identity.

Relationships between the Q-functions  are also known as Jacobi identities:

(2)

equivalent to the Jacobi triple product (Borwein and Borwein 1987, p. 65) and

(3)

where

(4)

 is the complete elliptic integral of the first kind, and . Using Weber functions

			(5)
			(6)
			(7)

(5) and (6) become

(8)

(9)

(Borwein and Borwein 1987, p. 69).

REFERENCES:

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, p. 3, 1996.