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 ${\rm{i}}K' \,$ are the quarter periods, and $\omega_1 \,$ and $\omega_2 \,$ are the fundamental pair of periods. Notationally, the quarter periods $K \,$ and ${\rm{i}}K' \,$ are usually used only in the context of the Jacobian elliptic functions, whereas the half-periods $\omega_1 \,$ and $\omega_2 \,$ are usually used only in the context of Weierstrass elliptic functions. Some authors, notably Apostol, use $\omega_1 \,$ and $\omega_2 \,$ 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 $\tau=\frac{{\rm{i}}K'}{K}=\frac{\omega_2}{\omega_1}\,$ is sometimes called the half-period ratio because it is the ratio of the two half-periods $\omega_1 \,$ and $\omega_2 \,$ of an elliptic function.

The complementary nome $q_1\,$ 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 0486612724
• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0