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