In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ... In mathematics, several functions are important enough to deserve their own name. ...

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. In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. ... In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ... In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ...

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. In mathematics, the half-period ratio τ of an elliptic function j is the ratio of the two half-periods ω1 and ω2 of j, where j is defined in such a way that See also Modular form Categories: Math stubs ...

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. In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler. ...

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