Function Repository Resource:

# EisensteinE

Evaluate the Eisenstein series

Contributed by: Jan Mangaldan
 ResourceFunction["EisensteinE"][n,q] gives the Eisenstein series En(q).

## Details and Options

Mathematical function, suitable for both symbolic and numerical manipulation.
ResourceFunction["EisensteinE"] is defined only for positive even n and inside the unit q-disk.
For certain special arguments, ResourceFunction["EisensteinE"] automatically evaluates to exact values.
ResourceFunction["EisensteinE"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (3)

Evaluate numerically:

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Plot E8(q) over a subset of the reals:

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Series expansion at the origin:

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### Scope (4)

Evaluate for complex arguments:

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Evaluate to high precision:

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Simple exact values are generated automatically:

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### Applications (1)

Plot near the unit circle in the complex q-plane:

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### Properties and Relations (2)

EisensteinE[n,q] for n>2 can be expressed in terms of EllipticTheta:

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WeierstrassInvariants can be expressed in terms of EisensteinE[4,q] and EisensteinE[6,q]:

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### Possible Issues (1)

EisensteinE[n,q] is only defined for positive even n:

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### Neat Examples (1)

Visualize a function with a boundary of analyticity:

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## Version History

• 1.0.0 – 09 December 2020