# EllipticTheta

EllipticTheta[a,u,q]

gives the theta function .

EllipticTheta[a,q]

gives the theta constant .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• .
• .
• .
• .
• The are defined only inside the unit q disk; the disk forms a natural boundary of analyticity.
• Inside the unit q disk, and have branch cuts from to .
• For certain special arguments, EllipticTheta automatically evaluates to exact values.
• EllipticTheta can be evaluated to arbitrary numerical precision.
• EllipticTheta automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin with respect to q:

## Scope(13)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(3)

Value at zero:

EllipticTheta evaluates symbolically for special arguments:

Find the first positive minimum of EllipticTheta[3,x,1/2]:

### Visualization(2)

Plot the EllipticTheta function for various parameters:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(4)

Real and complex domains of EllipticTheta:

EllipticTheta is a periodic function with respect to :

## Generalizations & Extensions(1)

EllipticTheta can be applied to a power series:

## Applications(7)

Plot near the unit circle in the complex q plane:

The number of representations of n as a sum of four squares:

Conformal map from an ellipse to the unit disk:

Visualize the map:

Dirichlet Green's function for the 1D heat equation:

Plot the timedependent temperature distribution:

Form Bloch functions of a onedimensional crystal with Gaussian orbitals:

Plot Bloch functions as a function of the quasiwave vector:

Electrostatic potential in a NaCllike crystal with point-like ions:

Plot the potential in a plane through the crystal:

A concise form of the Poisson summation formula:

## Properties & Relations(2)

Numerically find a root of a transcendental equation:

Sum can generate elliptic theta functions:

## Possible Issues(3)

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

Use arbitrary-precision arithmetic to obtain the correct result:

EllipticTheta has the attribute NHoldFirst:

Different argument conventions exist:

## Neat Examples(1)

Visualize a function with a boundary of analyticity: