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

q = exp( - πK' / K) = exp(iπω₂ / ω₁) = exp(iπτ)

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 = ω₂ / ω₁ 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 complimentary nome q₁ is given by

q = exp( - πK / K').

See the pages 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. ISBN 486-61272-4 . See sections 16.27.4 and 17.3.17
• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0