Evaluate numerically:

Plot over a subset of the reals:

Series expansion at the origin with respect to q:

EllipticTheta can be applied to a power series:

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 time‐dependent temperature distribution:

Form Bloch functions of a one‐dimensional crystal with Gaussian orbitals:

Plot Bloch functions as a function of the quasi‐wave vector:

Electrostatic potential in a NaCl‐like 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 arithmetic‐geometric 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:

Numerically find a root of a transcendental equation:

Sum can generate elliptic theta functions:

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: