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Inverse Nome


InverseNomeInverseNomeReIm

Solving the nome q for the parameter m gives

m(q)=(theta_2^4(q))/(theta_3^4(q))
(1)
=(16eta^8(1/2tau)eta^(16)(2tau))/(eta^(24)(tau)),
(2)

where theta_i(q)=theta_i(0,q) is a Jacobi theta function, eta(tau) is the Dedekind eta function, and q=e^(ipitau) is the nome.

The inverse nome function is essentially the same as the elliptic lambda function, the difference being that elliptic lambda function is a function of the half-period ratio tau, while the inverse nome is a function of the nome q, where q is itself a function of tau.

The inverse nome is implemented as InverseEllipticNomeQ[q] in the Wolfram Language.

As a rule, inverse and direct functions satisfy the relation f(f^(-1)(z))=z-for example, sin(sin^(-1)(z))=z. The inverse nome is an exception to this rule due to a historical mistake made more a century ago. In particular, the inverse nome and nome itself are connected by the opposite relation q^(-1)(q(m))=m.

Special values include

m(0)=0
(3)
m(e^pi)=1/2
(4)
m(1)=1,
(5)

although strictly speaking, q^(-1)(1) is not defined at 1 because q^(-1)(z) is a modular function, therefore has a dense set of singularities on the unit circle, and is therefore only defined strictly inside the unit circle.

It has series

 m(q)=16q-128q^2+704q^3-3072q^4+...
(6)

(OEIS A115977).

It satisfies

 lim_(q->0^+)(dm)/(dq)=16.
(7)

See also

Elliptic Lambda Function, Jacobi Theta Functions, Nome

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/InverseEllipticNomeQ/

Explore with Wolfram|Alpha

References

Sloane, N. J. A. Sequence A115977 in "The On-Line Encyclopedia of Integer Sequences."Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 66 and 89, 1999.

Referenced on Wolfram|Alpha

Inverse Nome

Cite this as:

Weisstein, Eric W. "Inverse Nome." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseNome.html

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