# 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.
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• The are only defined within the unit q disk, ; the unit disk forms a natural boundary of analyticity.
• Within 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.
• EllipticTheta can be used with Interval and CenteredInterval objects. »

# 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(20)

### Numerical Evaluation(5)

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:

EllipticTheta can be used with Interval and CenteredInterval objects:

### 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(10)

Real and complex domains of EllipticTheta:

EllipticTheta is a periodic function with respect to : is an analytic function of x:

For example, has no singularities or discontinuities: is neither nondecreasing nor nonincreasing: is not injective: is not surjective: is neither non-negative nor non-positive: is neither convex nor concave:

## Generalizations & Extensions(1)

EllipticTheta can be applied to a power series:

## Applications(11)

Plot theta functions near the unit circle in the complex q plane:

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

Verify Jacobi's triple product identity through a series expansion:

Conformal map from an ellipse to the unit disk:

Visualize the map:

Green's function for the 1D heat equation with Dirichlet boundary conditions and initial condition :

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:

An asymptotic approximation for a finite Gaussian sum:

Compare the approximate and exact values for :

Closed form of iterates of the arithmeticgeometric mean:

Compare the closed form with explicit iterations:

Form any elliptic function with given periods, poles and zeros as a rational function of EllipticTheta:

Form an elliptic function with a single and a double zero and a triple pole:

Plot the resulting elliptic function:

## Properties & Relations(2)

Numerically find a root of a transcendental equation:

Sum can generate elliptic theta functions:

## Possible Issues(4)

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

Use arbitrary-precision arithmetic to obtain the correct result:

The first argument must be an explicit integer between 1 and 4:

EllipticTheta has the attribute NHoldFirst:

Different argument conventions exist for theta functions:

## Neat Examples(1)

Visualize a function with a boundary of analyticity: