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EllipticThetaPrime
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EllipticThetaPrime

✖

gives the derivative with respect to u of the theta function TemplateBox[{a, u, q}, EllipticTheta] .

✖

gives the theta constant TemplateBox[{a, 0, q}, EllipticThetaPrime].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For certain special arguments, EllipticThetaPrime automatically evaluates to exact values.
  • EllipticThetaPrime can be evaluated to arbitrary numerical precision.
  • EllipticThetaPrime automatically threads over lists.

Examples

open allclose all

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

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-e5qhru
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-ckr0fe
Out[1]=1

Series expansion at the origin with respect to q:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-rg9gu
Out[1]=1

Scope  (21)Survey of the scope of standard use cases

Numerical Evaluation  (4)

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-l274ju
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-cksbl4
Out[2]=2

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-b0wt9
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-y7k4a
Out[2]=2

Complex number inputs:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-hfml09
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-bq2c6r
Out[2]=2

Specific Values  (3)

Value at zero:

In[1]:=1
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https://wolfram.com/xid/0yv98cp326a8-cgkwdk
Out[1]=1

EllipticThetaPrime evaluates symbolically for special arguments:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-gv2gig
Out[1]=1

Find a value of for which EllipticThetaPrime[3,x,1/2]=2:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-f2hrld
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-emazsg
Out[2]=2

Visualization  (2)

Plot the EllipticThetaPrime function for various parameters:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-drb1t8
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-b3jtoi
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-hr6dbn
Out[2]=2

Function Properties  (10)

Real and complex domains of EllipticThetaPrime:

In[1]:=1
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https://wolfram.com/xid/0yv98cp326a8-cl7ele
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0yv98cp326a8-de3irc
Out[2]=2

EllipticThetaPrime is a periodic function with respect to :

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-bx99ar
Out[1]=1

EllipticThetaPrime threads elementwise over lists:

In[1]:=1
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https://wolfram.com/xid/0yv98cp326a8-bbdzyd
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0yv98cp326a8-3tafr
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-h5x4l2
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0yv98cp326a8-mdtl3h
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0yv98cp326a8-mn5jws
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-nlz7s
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0yv98cp326a8-poz8g
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0yv98cp326a8-ctca0g
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-cxk3a6
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0yv98cp326a8-frlnsr
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0yv98cp326a8-84dui
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-8kku21
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-c4luor
In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-dgveza

Integration  (2)

Compute the indefinite integral using Integrate:

In[1]:=1
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https://wolfram.com/xid/0yv98cp326a8-bponid
Out[1]=1

Verify the anti-derivative:

In[2]:=2
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https://wolfram.com/xid/0yv98cp326a8-op9yly
Out[2]=2

Definite integral:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-bfdh5d
Out[1]=1

Generalizations & Extensions  (1)Generalized and extended use cases

EllipticThetaPrime can be applied to power series:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-ce4ga
Out[1]=1

Applications  (4)Sample problems that can be solved with this function

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

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-zv0i6
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0yv98cp326a8-mhlrm

Calculate the temperature gradient:

In[2]:=2
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https://wolfram.com/xid/0yv98cp326a8-35ujw
Out[2]=2

Plot the temperature gradient:

In[3]:=3
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https://wolfram.com/xid/0yv98cp326a8-c0k8dc
Out[3]=3

Electrostatic force in a NaCl‐like crystal with point‐like ions:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-g5par3

Plot the magnitude of the force in a plane through the crystal:

In[2]:=2
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https://wolfram.com/xid/0yv98cp326a8-e504x2
Out[2]=2

The canonical rotational distribution function for linear molecules :

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-gmxkr8

Plot a numerical approximation of the partition function:

In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-jjnz8
Out[2]=2

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/0yv98cp326a8-fzmuhx
Out[1]=1

Use arbitrary-precision arithmetic to obtain the correct result:

In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-ei7nal
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-bwp2ya
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-2w91a
Out[2]=2

EllipticThetaPrime has the attribute NHoldFirst:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-fn8uon
Out[1]=1

Different argument conventions exist for theta functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv98cp326a8-k4z8az
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv98cp326a8-c071s9
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/0yv98cp326a8-bimsst
Out[1]=1
Wolfram Research (1996), EllipticThetaPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticThetaPrime.html (updated 2017).
✖
Wolfram Research (1996), EllipticThetaPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticThetaPrime.html (updated 2017).

Text

Wolfram Research (1996), EllipticThetaPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticThetaPrime.html (updated 2017).

✖
Wolfram Research (1996), EllipticThetaPrime, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticThetaPrime.html (updated 2017).

CMS

Wolfram Language. 1996. "EllipticThetaPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/EllipticThetaPrime.html.

✖
Wolfram Language. 1996. "EllipticThetaPrime." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/EllipticThetaPrime.html.

APA

Wolfram Language. (1996). EllipticThetaPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EllipticThetaPrime.html

✖
Wolfram Language. (1996). EllipticThetaPrime. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EllipticThetaPrime.html

BibTeX

@misc{reference.wolfram_2025_ellipticthetaprime, author="Wolfram Research", title="{EllipticThetaPrime}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticThetaPrime.html}", note=[Accessed: 26-March-2025 ]}

✖
@misc{reference.wolfram_2025_ellipticthetaprime, author="Wolfram Research", title="{EllipticThetaPrime}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticThetaPrime.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_ellipticthetaprime, organization={Wolfram Research}, title={EllipticThetaPrime}, year={2017}, url={https://reference.wolfram.com/language/ref/EllipticThetaPrime.html}, note=[Accessed: 26-March-2025 ]}

✖
@online{reference.wolfram_2025_ellipticthetaprime, organization={Wolfram Research}, title={EllipticThetaPrime}, year={2017}, url={https://reference.wolfram.com/language/ref/EllipticThetaPrime.html}, note=[Accessed: 26-March-2025 ]}