DedekindEta

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`DedekindEta`

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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 half‐periods .
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.
DedekindEta can be used with CenteredInterval objects. »

Examples

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Basic Examples (2)Summary of the most common use cases

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-c4ku41

Out[1]=1

Plot over a subset of the reals:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-dw6o47

Out[1]=1

Scope (14)Survey of the scope of standard use cases

Numerical Evaluation (4)

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-l274ju

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-cksbl4

Out[2]=2

Evaluate to high precision:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-b0wt9

Out[1]=1

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

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-y7k4a

Out[2]=2

Evaluate efficiently at high precision:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-di5gcr

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-bq2c6r

Out[2]=2

DedekindEta can be used with CenteredInterval objects:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-cmdnbi

Out[1]=1

Specific Values (2)

Value at fixed point:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-nww7l

Out[1]=1

DedekindEta threads elementwise over lists:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-pj5yw

Out[1]=1

Visualization (2)

Plot the DedekindEta function for various parameters:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-drb1t8

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-i75zi3

Out[1]=1

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

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-ij0b0x

Out[2]=2

Function Properties (6)

Complex domain of DedekindEta:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-cl7ele

Out[1]=1

DedekindEta is a periodic function:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-bx99ar

Out[1]=1

DedekindEta is an analytic function on its domain:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-wtkikd

Out[1]=1

It is not an entire function, however:

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-h5x4l2

Out[2]=2

It has both singularities and discontinuities:

In[3]:=3

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https://wolfram.com/xid/0ywcubs65tnf-mdtl3h

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0ywcubs65tnf-mn5jws

Out[4]=4

DedekindEta is not injective over the complexes:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-poz8g

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-ctca0g

Out[2]=2

DedekindEta is not surjective:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-cxk3a6

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-frlnsr

Out[2]=2

TraditionalForm formatting:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-mre0k

Applications (3)Sample problems that can be solved with this function

The modular discriminant at I is given by DedekindEta:

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-lr2vd

Out[2]=2

Compare with the general definition:

In[4]:=4

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https://wolfram.com/xid/0ywcubs65tnf-zkdlz

Out[5]=5

Plot the DedekindEta function in the upper half of the complex plane:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-m3r9fq

Out[1]=1

The modular discriminant:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-8v5zw0

Relation with DedekindEta:

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-mwacl5

Out[2]=2

Properties & Relations (2)Properties of the function, and connections to other functions

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

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-bhiqxx

Out[1]=1

With exact input, the answer is correct:

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-hurk9v

Out[2]=2

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:

In[1]:=1

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https://wolfram.com/xid/0ywcubs65tnf-d8xpgj

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywcubs65tnf-lsbdzd

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0ywcubs65tnf-c84nd8

Out[3]=3

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