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EllipticTheta
✖
EllipticTheta

✖
EllipticTheta[a,u,q]

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

✖

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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-9427w
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-js5hl
Out[1]=1

Series expansion at the origin with respect to q:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-jzg97r
Out[1]=1

Scope  (20)Survey of the scope of standard use cases

Numerical Evaluation  (5)

Evaluate numerically:

In[1]:=1
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https://wolfram.com/xid/0b8day5o9i-l274ju
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8day5o9i-cksbl4
Out[2]=2

Evaluate to high precision:

In[1]:=1
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https://wolfram.com/xid/0b8day5o9i-b0wt9
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0b8day5o9i-y7k4a
Out[2]=2

Complex number inputs:

In[1]:=1
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https://wolfram.com/xid/0b8day5o9i-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-di5gcr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8day5o9i-bq2c6r
Out[2]=2

EllipticTheta can be used with Interval and CenteredInterval objects:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-psgutd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-enzbe7
Out[2]=2

Specific Values  (3)

Value at zero:

In[1]:=1
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https://wolfram.com/xid/0b8day5o9i-cgkwdk
Out[1]=1

EllipticTheta evaluates symbolically for special arguments:

In[1]:=1
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https://wolfram.com/xid/0b8day5o9i-bdag4x
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8day5o9i-bo964q
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0b8day5o9i-f2hrld
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-dd7na2
Out[2]=2

Visualization  (2)

Plot the EllipticTheta function for various parameters:

In[1]:=1
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https://wolfram.com/xid/0b8day5o9i-drb1t8
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0b8day5o9i-c7sf4v
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0b8day5o9i-xul6k
Out[2]=2

Function Properties  (10)

Real and complex domains of EllipticTheta:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-5t0713
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8day5o9i-de3irc
Out[2]=2

EllipticTheta is a periodic function with respect to :

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-bx99ar
Out[1]=1

EllipticTheta threads elementwise over lists:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-g3cl5t
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8day5o9i-be4iot
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-s9ypiy
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0b8day5o9i-mdtl3h
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0b8day5o9i-8alhkv
In[4]:=4
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https://wolfram.com/xid/0b8day5o9i-mn5jws
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-nlz7s
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-poz8g
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b8day5o9i-ctca0g
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-cxk3a6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-frlnsr
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-84dui
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-8kku21
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-glpjv1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-qfs8b

Generalizations & Extensions  (1)Generalized and extended use cases

EllipticTheta can be applied to a power series:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i
Out[1]=1

Applications  (11)Sample problems that can be solved with this function

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-066ah
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-ckybw
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-bkguon
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-zv0i6
Out[1]=1

Conformal map from an ellipse to the unit disk:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i

Visualize the map:

In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i

Plot the time‐dependent temperature distribution:

In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-jjafwo
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i

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

In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i
Out[2]=2

Electrostatic potential in a NaCl‐like crystal with point-like ions:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-pevwn
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i

Plot the potential in a plane through the crystal:

In[3]:=3
✖
https://wolfram.com/xid/0b8day5o9i
Out[3]=3

A concise form of the Poisson summation formula:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i
Out[2]=2

An asymptotic approximation for a finite Gaussian sum:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-cdpj8m
Out[1]=1

Compare the approximate and exact values for :

In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-clu3dc
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8day5o9i-fk652r
Out[3]=3

Closed form of iterates of the arithmetic‐geometric mean:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-dr56ed

Compare the closed form with explicit iterations:

In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-ci0txa
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b8day5o9i-cyv4dy
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-dmv8tz

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

In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-c9vhi1
Out[2]=2

Plot the resulting elliptic function:

In[3]:=3
✖
https://wolfram.com/xid/0b8day5o9i-c52bdi
Out[3]=3

Properties & Relations  (2)Properties of the function, and connections to other functions

Numerically find a root of a transcendental equation:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i
Out[1]=1

Sum can generate elliptic theta functions:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i
Out[1]=1

Possible Issues  (4)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i
Out[1]=1

Use arbitrary-precision arithmetic to obtain the correct result:

In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-bwp2ya
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i-2w91a
Out[2]=2

EllipticTheta has the attribute NHoldFirst:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i
Out[1]=1

Different argument conventions exist for theta functions:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Visualize a function with a boundary of analyticity:

In[1]:=1
✖
https://wolfram.com/xid/0b8day5o9i-bnj6fk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b8day5o9i
Out[2]=2
Wolfram Research (1988), EllipticTheta, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticTheta.html (updated 2022).
✖
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).

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_elliptictheta, author="Wolfram Research", title="{EllipticTheta}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticTheta.html}", note=[Accessed: 08-July-2025 ]}

✖
@misc{reference.wolfram_2025_elliptictheta, author="Wolfram Research", title="{EllipticTheta}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticTheta.html}", note=[Accessed: 08-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_elliptictheta, organization={Wolfram Research}, title={EllipticTheta}, year={2022}, url={https://reference.wolfram.com/language/ref/EllipticTheta.html}, note=[Accessed: 08-July-2025 ]}

✖
@online{reference.wolfram_2025_elliptictheta, organization={Wolfram Research}, title={EllipticTheta}, year={2022}, url={https://reference.wolfram.com/language/ref/EllipticTheta.html}, note=[Accessed: 08-July-2025 ]}