EllipticThetaPrime

EllipticThetaPrime[a,u,q]

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

EllipticThetaPrime[a,q]

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

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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  (21)

Numerical Evaluation  (4)

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:

Specific Values  (3)

Value at zero:

EllipticThetaPrime evaluates symbolically for special arguments:

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

Visualization  (2)

Plot the EllipticThetaPrime function for various parameters:

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

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

Function Properties  (10)

Real and complex domains of EllipticThetaPrime:

EllipticThetaPrime is a periodic function with respect to :

EllipticThetaPrime threads elementwise over lists:

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

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

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

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

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

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

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

TraditionalForm formatting:

Integration  (2)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

Generalizations & Extensions  (1)

EllipticThetaPrime can be applied to power series:

Applications  (4)

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

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

Calculate the temperature gradient:

Plot the temperature gradient:

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

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

The canonical rotational distribution function for linear molecules :

Plot a numerical approximation of the partition function:

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:

EllipticThetaPrime has the attribute NHoldFirst:

Different argument conventions exist for theta functions:

Neat Examples  (1)

Visualize a function with a boundary of analyticity:

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).

CMS

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

BibTeX

@misc{reference.wolfram_2024_ellipticthetaprime, author="Wolfram Research", title="{EllipticThetaPrime}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticThetaPrime.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_ellipticthetaprime, organization={Wolfram Research}, title={EllipticThetaPrime}, year={2017}, url={https://reference.wolfram.com/language/ref/EllipticThetaPrime.html}, note=[Accessed: 21-December-2024 ]}