DedekindEta

DedekindEta[τ]

gives the Dedekind eta modular elliptic function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • DedekindEta is defined only in the upper half of the complex τ plane. It is not defined for real τ.
  • The argument τ is the ratio of Weierstrass halfperiods .
  • DedekindEta satisfies where is the discriminant, given in terms of Weierstrass invariants by .
  • For certain special arguments, DedekindEta automatically evaluates to exact values.
  • DedekindEta can be evaluated to arbitrary numerical precision.
  • DedekindEta automatically threads over lists.

Examples

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Basic Examples  (2)

Evaluate numerically:

Plot over a subset of the reals:

Scope  (10)

Numerical Evaluation  (3)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Specific Values  (2)

Value at fixed point:

DedekindEta threads elementwise over lists:

Visualization  (2)

Plot the DedekindEta function for various parameters:

Plot the real part of the DedekindEta function in three dimensions:

Plot the imaginary part of the DedekindEta function in three dimensions:

Function Properties  (3)

Complex domain of DedekindEta:

DedekindEta is a periodic function:

TraditionalForm formatting:

Applications  (2)

Plot the DedekindEta in the upperhalf complex plane:

The modular discriminant:

Relation with DedekindEta:

Properties & Relations  (2)

Machine-precision input is insufficient to give a correct answer:

With exact input, the answer is correct:

Because DedekindEta is a numerical function with numeric arguments, it might be considered a numeric quantity but because of its boundary of analyticity, it might not be evaluatable to a number:

Introduced in 1996
 (3.0)