Nome (mathematics)

In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by

$q =e^{-\frac{\pi K'}{K}} =e^{\frac{{\rm{i}}\pi\omega_2}{\omega_1}} =e^{{\rm{i}} \pi \tau} \,$

where K and iK ′ are the quarter periods, and ω₁ and ω₂ are the fundamental pair of periods. Notationally, the quarter periods K and iK ′ are usually used only in the context of the Jacobian elliptic functions, whereas the half-periods ω₁ and ω₂ are usually used only in the context of Weierstrass elliptic functions. Some authors, notably Apostol, use ω₁ and ω₂ to denote whole periods rather than half-periods.

The nome is frequently used as a value with which elliptic functions and modular forms can be described; on the other hand, it can also be thought of as function, because the quarter periods are functions of the elliptic modulus. This ambiguity occurs because for real values of the elliptic modulus, the quarter periods and thus the nome are uniquely determined.

The function τ = iK ′/K = ω₁/ω₂ is sometimes called the half-period ratio because it is the ratio of the two half-periods ω₁ and ω₂ of an elliptic function.

The complementary nome q₁ is given by

$q_1=e^{-\frac{\pi K}{K'}}. \,$

See the articles on quarter period and elliptic integrals for additional definitions and relations on the nome.

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. OCLC 1097832 . See sections 16.27.4 and 17.3.17. 1972 edition: ISBN 0-486-61272-4
• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0