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q=e^{-\pi K'/K}=e^{-i\pi\tau},
\end{displaymath} (1)

$\displaystyle Q_0$ $\textstyle \equiv$ $\displaystyle \prod_{n=1}^\infty (1-q^{2n})$ (2)
$\displaystyle Q_1$ $\textstyle \equiv$ $\displaystyle \prod_{n=1}^\infty (1+q^{2n})$ (3)
$\displaystyle Q_2$ $\textstyle \equiv$ $\displaystyle \prod_{n=1}^\infty (1+q^{2n-1})$ (4)
$\displaystyle Q_3$ $\textstyle \equiv$ $\displaystyle \prod_{n=1}^\infty (1-q^{2n-1}).$ (5)

The $Q$-functions are sometimes written using a lower-case $q$ instead of a capital $Q$. The $Q$-functions also satisfy the identities
$\displaystyle Q_0Q_1$ $\textstyle =$ $\displaystyle Q_0(q^2)$ (6)
$\displaystyle Q_0Q_3$ $\textstyle =$ $\displaystyle Q_0(q^{1/2})$ (7)
$\displaystyle Q_2Q_3$ $\textstyle =$ $\displaystyle Q_3(q^2)$ (8)
$\displaystyle Q_1Q_2$ $\textstyle =$ $\displaystyle Q_1(q^{1/2}).$ (9)

See also Jacobi Identities, q-Series


Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 55 and 63-85, 1987.

Tannery, J. and Molk, J. Elements de la Théorie des Fonctions Elliptiques, 4 vols. Paris: Gauthier-Villars et fils, 1893-1902.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 469-473 and 488-489, 1990.

© 1996-9 Eric W. Weisstein