gives the theta function .
gives the theta constant .
- 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.
- EllipticTheta can be used with Interval and CenteredInterval objects. »
Examplesopen allclose all
Basic Examples (3)
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
EllipticTheta can be used with Interval and CenteredInterval objects:
Specific Values (3)
EllipticTheta evaluates symbolically for special arguments:
Find the first positive minimum of EllipticTheta[3,x,1/2]:
Plot the EllipticTheta function for various parameters:
Function Properties (10)
Real and complex domains of EllipticTheta:
EllipticTheta is a periodic function with respect to :
EllipticTheta threads elementwise over lists:
For example, has no singularities or discontinuities:
is neither nondecreasing nor nonincreasing:
is neither non-negative nor non-positive:
is neither convex nor concave:
Generalizations & Extensions (1)
EllipticTheta can be applied to a power series:
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 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:
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:
Wolfram Research (1988), EllipticTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticTheta.html (updated 2022).
Wolfram Language. 1988. "EllipticTheta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/EllipticTheta.html.
Wolfram Language. (1988). EllipticTheta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EllipticTheta.html