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# Klein's Absolute Invariant

Min Max
 Min Max Re Im

Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as

 (1)

where and are the invariants of the Weierstrass elliptic function with modular discriminant

 (2)

(Klein 1877). If , where is the upper half-plane, then

 (3)

is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in (Apostol 1997, p. 15).

Klein's absolute invariant is implemented in the Wolfram Language as KleinInvariantJ[tau].

The function is the same as the j-function, modulo a constant multiplicative factor.

Every rational function of is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40).

Klein's invariant can be given explicitly by

 (4) (5)

(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function

 (6)

is a Jacobi theta function, the are Eisenstein series, and is the nome. Klein's invariant can also be simply expressed in terms of the five Weber functions , , , , and .

is invariant under a unimodular transformation, so

 (7)

and is a modular function. takes on the special values

 (8) (9) (10)

satisfies the functional equations

 (11) (12)

It satisfies a number of beautiful multiple-argument identities, including the duplication formula

 (13) (14)

with

 (15) (16)

and the Dedekind eta function, the triplication formula

 (17) (18)

with

 (19) (20)

and the quintuplication formula

 (21) (22)

with

 (23) (24)

Plotting the real or imaginary part of in the complex plane produces a beautiful fractal-like structure, illustrated above.

## See also

Elliptic Lambda Function, Eisenstein Series, j-Function, Jacobi Theta Functions, Pi, Weber Functions

## Related Wolfram sites

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

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## References

Apostol, T. M. "Klein's Modular Function ," "Invariance of Under Unimodular Transformation," "The Fourier Expansions of and ," "Special Values of ," and "Modular Functions as Rational Functions of ." §1.12-1.13, 1.15, and 2.5-2.6 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 15-18, 20-22, and 39-40, 1997.Brezhnev, Y. V. "Uniformisation: On the Burnside Curve ." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 115 and 179, 1987.Cohn, H. Introduction to the Construction of Class Fields. New York: Dover, p. 73, 1994.Klein, F. "Sull' equazioni dell' Icosaedro nella risoluzione delle equazioni del quinto grado [per funzioni ellittiche]." Reale Istituto Lombardo, Rendiconto, Ser. 2 10, 1877.Klein, F. "Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades." Math. Ann. 14, 111-172, 1878-1879.Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. Unpublished manuscript. 1999.

## Referenced on Wolfram|Alpha

Klein's Absolute Invariant

## Cite this as:

Weisstein, Eric W. "Klein's Absolute Invariant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KleinsAbsoluteInvariant.html