EllipticTheta

EllipticTheta[a,u,q]

gives the theta function TemplateBox[{a, u, q}, EllipticTheta] .

EllipticTheta[a,q]

gives the theta constant TemplateBox[{a, q}, EllipticThetaConstant]=TemplateBox[{a, 0, q}, EllipticTheta].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{1, u, q}, EllipticTheta]=2q^(1/4)sum_(n=0)^(infty)(-1)^nq^(n(n+1))sin((2 n+1)u).
  • TemplateBox[{2, u, q}, EllipticTheta]=2q^(1/4)sum_(n=0)^(infty)q^(n(n+1))cos((2 n+1)u).
  • TemplateBox[{3, u, q}, EllipticTheta]=1+2sum_(n=1)^(infty)q^(n^2)cos(2n u).
  • TemplateBox[{4, u, q}, EllipticTheta]=1+2sum_(n=1)^(infty)(-1)^nq^(n^2)cos(2n u).
  • The are only defined within the unit q disk, TemplateBox[{q}, Abs]<1; 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 TemplateBox[{4, z, {1, /, 3}}, EllipticTheta]:

Plot the imaginary part of TemplateBox[{4, z, {1, /, 3}}, EllipticTheta]:

Function Properties  (10)

Real and complex domains of EllipticTheta:

EllipticTheta is a periodic function with respect to :

EllipticTheta threads elementwise over lists:

TemplateBox[{1, x, q}, EllipticTheta] is an analytic function of x:

For example, TemplateBox[{1, x, {1, /, 2}}, EllipticTheta] has no singularities or discontinuities:

TemplateBox[{1, x, {1, /, 2}}, EllipticTheta] is neither nondecreasing nor nonincreasing:

TemplateBox[{1, x, {1, /, 2}}, EllipticTheta] is not injective:

TemplateBox[{1, x, {1, /, 2}}, EllipticTheta] is not surjective:

TemplateBox[{1, x, {1, /, 2}}, EllipticTheta] is neither non-negative nor non-positive:

TemplateBox[{1, x, {1, /, 2}}, EllipticTheta] is neither convex nor concave:

TraditionalForm formatting:

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:

Wolfram Research (1988), EllipticTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticTheta.html (updated 2022).

Text

Wolfram Research (1988), EllipticTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticTheta.html (updated 2022).

CMS

Wolfram Language. 1988. "EllipticTheta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/EllipticTheta.html.

APA

Wolfram Language. (1988). EllipticTheta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EllipticTheta.html

BibTeX

@misc{reference.wolfram_2023_elliptictheta, author="Wolfram Research", title="{EllipticTheta}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticTheta.html}", note=[Accessed: 19-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_elliptictheta, organization={Wolfram Research}, title={EllipticTheta}, year={2022}, url={https://reference.wolfram.com/language/ref/EllipticTheta.html}, note=[Accessed: 19-March-2024 ]}