JacobiZN

JacobiZN[u,m]

gives the Jacobi zeta function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • zn(u|m)=int_0^u(TemplateBox[{z, m}, JacobiDN]^2-TemplateBox[{m}, EllipticE]/TemplateBox[{m}, EllipticK])dz.
  • Argument conventions for elliptic integrals are discussed in "Elliptic Integrals and Elliptic Functions".
  • is a singly periodic function in with the period 2 TemplateBox[{m}, EllipticK], where is the elliptic integral EllipticK. »
  • JacobiZN is a meromorphic function in both arguments.
  • For certain special arguments, JacobiZN automatically evaluates to exact values.
  • JacobiZN can be evaluated to arbitrary numerical precision.
  • JacobiZN automatically threads over lists.

Examples

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Basic Examples  (3)

Evaluate numerically:

Series expansion about the origin:

Scope  (23)

Numerical Evaluation  (4)

Evaluate to high precision:

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

Evaluate for complex arguments:

Evaluate JacobiZN efficiently at higher precision:

JacobiZN threads elementwise over lists:

Specific Values  (3)

Simple exact values are generated automatically:

JacobiZN has poles coinciding with poles of JacobiDN:

Find a root of JacobiZN[u,2/3]=1/7:

Visualization  (3)

Plot JacobiZN functions for various values of parameter m:

Plot JacobiZN as a function of its parameter m:

Plot the real part of JacobiZN[x+y,1/2]:

Plot the imaginary part of JacobiZN[x+y,1/2]:

Function Properties  (2)

JacobiZN is periodic with period 2 TemplateBox[{m}, EllipticK]:

JacobiZN is additive quasiperiodic with a quasiperiod of 2 ⅈ TemplateBox[{{1, -, m}}, EllipticK]:

JacobiZN is an odd function:

Differentiation  (3)

First derivative:

Higher-order derivatives:

Plot derivatives for parameter :

Derivative with respect to parameter m:

Integration  (1)

Indefinite integral of JacobiZN:

Series Expansions  (3)

Series expansion for JacobiZN[u,1/3] around :

Plot three approximations for JacobiZN[u,1/3]:

Taylor series for JacobiZN[2,m] around :

Plot series approximations for JacobiZN[2,m]:

JacobiZN can be applied to power series:

Function Identities and Simplifications  (2)

Parity transformation and quasiperiodicity relations are automatically applied:

Automatic argument simplification:

Function Representations  (2)

JacobiZN is related to JacobiZeta function:

TraditionalForm formatting:

Applications  (4)

Express derivatives of Neville theta functions:

Supersymmetric zero‐energy solution of the Schrödinger equation in a periodic potential:

Define a solution using JacobiZN:

Check that the function defined previously solves the Schrödinger equation:

Plot the superpotential, the potential and the wavefunction:

Define a conformal map using JacobiZN:

Parameterization of genus‐1 constant mean-curvature Wente torus:

Visualize 3‐lobe, 5‐lobe, 7‐lobe and 11‐lobe tori:

Properties & Relations  (2)

JacobiZN is defined in terms of JacobiEpsilon:

JacobiZN[u,m] is a meromorphic extension of TemplateBox[{TemplateBox[{u, m}, JacobiAmplitude], m}, JacobiZeta]:

Wolfram Research (2020), JacobiZN, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiZN.html.

Text

Wolfram Research (2020), JacobiZN, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiZN.html.

CMS

Wolfram Language. 2020. "JacobiZN." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiZN.html.

APA

Wolfram Language. (2020). JacobiZN. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiZN.html

BibTeX

@misc{reference.wolfram_2023_jacobizn, author="Wolfram Research", title="{JacobiZN}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiZN.html}", note=[Accessed: 04-October-2023 ]}

BibLaTeX

@online{reference.wolfram_2023_jacobizn, organization={Wolfram Research}, title={JacobiZN}, year={2020}, url={https://reference.wolfram.com/language/ref/JacobiZN.html}, note=[Accessed: 04-October-2023 ]}