Given a Jacobi theta function, the nome is defined as


(Borwein and Borwein 1987, pp. 41, 109 and 114), where tau is the half-period ratio, K(k) is the complete elliptic integral of the first kind, and k is the elliptic modulus. The nome is implemented in the Wolfram Language as EllipticNomeQ[m].

Extreme care is needed when consulting the literature, as it is common in the theory of modular functions (and in particular the Dedekind eta function) to use the symbol q to denote e^(2piitau), i.e., the square of the usual nome (e.g., Berndt 1993, p. 139). In this work, the modular version of q is denoted


(e.g., Borwein and Borwein 1987, p. 118).


The nome in plotted above in the complex k-plane.

Various notations for Jacobi theta functions involving the nome include


where tau is the half-period ratio (Whittaker and Watson 1990, p. 464) and


Special values include


The nome has Maclaurin series in parameter m given by


(OEIS A002639 and A119349).

The nome has derivative


where K(m) is a complete elliptic integral of the first kind and m=k^2 is the elliptic modulus.

There exists a nonlinear third-order differential equation


for q(m) (Bertrand and Zudilin 2000; Trott 2006, pp. 29-31).

See also

Elliptic Characteristic, Elliptic Integral, Elliptic Modulus, Half-Period Ratio, Inverse Nome, Jacobi Amplitude, Jacobi Theta Functions, Modular Angle, Modular Discriminant, Parameter

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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 591, 1972.Berndt, B. C. Ramanujan's Theory of Theta-Functions, Theta Functions: from the Classical to the Modern. Providence, RI: Amer. Math. Soc., pp. 1-63, 1993.Bertrand, D.; and Zudilin, W. "On the Transcendence Degree of the Differential Field Generated by Siegel Modular Forms." 23 Jun 2000., J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Bramhall, J. N. "An Iterative Method for Inversion of Power Series." Comm. ACM 4, 317-318, 1961.Ferguson, H. R. P.; Nielsen, D. E.; and Cook, G. "A Partition Formula for the Integer Coefficients of the Theta Function Nome." Math. Comput. 29, 851-855, 1975.Fettis, H. E. "Note on the Computation of Jacobi's Nome and Its Inverse." Computing 4, 202-206, 1969.Fletcher, A. §III in "Guide to Tables of Elliptic Functions." Math. Tables Other Aids Computation 3, 229-281, 1948."Guide to Tables." §III in Math. Tables Other Aids Computation 3, 234, 1948.Hermite, C. Oeuvres, Vol. 4. Paris: Gauthier-Villars, p. 477, 1917.Lowan, A. N.; Blanch, G.; and Horenstein, W. "On the Inversion of the q-Series Associated with Jacobian Elliptic Functions." Bull. Amer. Math. Soc. 48, 737-738, 1942.Sloane, N. J. A. Sequences A002639/M5108 and A119349 in "The On-Line Encyclopedia of Integer Sequences."Tannery, J. and Molk, J. Eléments de la théorie des fonctions elliptiques, Vol. 4. Paris: Gauthier-Villars, p. 141, 1902.Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006., Z. X. and Guo, D. R. Special Functions. Singapore: World Scientific, p. 512, 1989.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

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Cite this as:

Weisstein, Eric W. "Nome." From MathWorld--A Wolfram Web Resource.

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