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Jacobi Identities

``The'' Jacobi identity is a relationship

\end{displaymath} (1)

between three elements $A$, $B$, and $C$, where $[A,B]$ is the Commutator. The elements of a Lie Group satisfy this identity.

Relationships between the Q-Function $Q_i$ are also known as Jacobi identities:

\end{displaymath} (2)

equivalent to the Jacobi Triple Product (Borwein and Borwein 1987, p. 65) and
\end{displaymath} (3)

q\equiv e^{-\pi K'(k)/K(k)},
\end{displaymath} (4)

$K=K(k)$ is the complete Elliptic Integral of the First Kind, and $K'(k)=K(k')=K(\sqrt{1-k^2}\,)$. Using Weber Functions
$\displaystyle f_1$ $\textstyle =$ $\displaystyle q^{-1/24} Q_3$ (5)
$\displaystyle f_2$ $\textstyle =$ $\displaystyle 2^{1/2} q^{1/12}Q_1$ (6)
$\displaystyle f$ $\textstyle =$ $\displaystyle q^{-1/24}Q_2,$ (7)

(5) and (6) become
\end{displaymath} (8)

\end{displaymath} (9)

(Borwein and Borwein 1987, p. 69).

See also Commutator, Jacobi Triple Product, Q-Function, Weber Functions


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

© 1996-9 Eric W. Weisstein