Inverse Nome


Solving the nome q for the parameter m gives


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


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


(OEIS A115977).

It satisfies


See also

Elliptic Lambda Function, Jacobi Theta Functions, Nome

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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.

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