JacobiEpsilon [u,m]
gives the Jacobi epsilon function TemplateBox[{u, m}, JacobiEpsilon].
JacobiEpsilon
JacobiEpsilon [u,m]
gives the Jacobi epsilon function TemplateBox[{u, m}, JacobiEpsilon].
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- TemplateBox[{u, m}, JacobiEpsilon]=int_0^uTemplateBox[{z, m}, JacobiDN]^2dz.
- Argument conventions for elliptic integrals are discussed in "Elliptic Integrals and Elliptic Functions".
- JacobiEpsilon is a meromorphic function in both arguments.
- For certain special arguments, JacobiEpsilon automatically evaluates to exact values.
- JacobiEpsilon can be evaluated to arbitrary numerical precision.
- JacobiEpsilon automatically threads over lists.
Examples
open all close allBasic 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 JacobiEpsilon efficiently at high precision:
JacobiEpsilon threads elementwise over lists:
Specific Values (3)
Simple exact values are generated automatically:
JacobiEpsilon has poles coinciding with the poles of JacobiDN :
Find a root of JacobiEpsilon [u,]=2:
Visualization (3)
Plot the JacobiEpsilon functions for various values of parameter m:
Plot JacobiEpsilon as a function of its parameter m:
Plot the real part of JacobiEpsilon [x+y,]:
Plot the imaginary part of JacobiEpsilon [x+y,]:
Function Properties (2)
JacobiEpsilon is additive quasiperiodic with quasiperiod 2 TemplateBox[{m}, EllipticK]:
JacobiEpsilon is additive quasiperiodic with quasiperiod 2 ⅈ TemplateBox[{{1, -, m}}, EllipticK]:
JacobiEpsilon 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 JacobiEpsilon :
Series Expansions (3)
Series expansion for JacobiEpsilon [u,]:
Plot the first three approximations for JacobiEpsilon [u,] around :
Taylor expansion for JacobiEpsilon [2,m]:
Plot the first three series approximations for JacobiEpsilon [2,m] around :
JacobiEpsilon can be applied to power series:
Function Identities and Simplifications (2)
Parity transformation and quasiperiodicity relations are automatically applied:
Automatic argument simplifications:
Function Representations (2)
JacobiEpsilon is related to the elliptic integral of the second kind:
TraditionalForm formatting:
Applications (7)
JacobiEpsilon arises in derivatives of Jacobi elliptic functions with respect to parameter :
Plot JacobiEpsilon over the complex plane:
Motion of a charged particle in a magnetic field:
Verify that it solves Newton's equation of motion with Lorentz force:
Plot particle trajectories for several different initial velocities:
Parameterization of a rotating elastic rod (fixed at the origin):
Plot the shape of the deformed rod:
The parameterization parameter is the length of the rod:
Parameterization of Costa's minimal surface [MathWorld]:
Parameterization of the Chen–Gackstatter minimal surface:
Construct nonperiodic solutions of the Lamé differential equation from periodic solutions:
Verify that they satisfy the Lamé equation:
Plot all the solutions together:
Properties & Relations (3)
JacobiEpsilon is defined as a definite integral of TemplateBox[{u, m}, JacobiDN]^2:
JacobiEpsilon [u,m] is a meromorphic extension of TemplateBox[{TemplateBox[{u, m}, JacobiAmplitude], m}, EllipticE2]:
JacobiEpsilon is related to JacobiZN :
See Also
Related Guides
History
Text
Wolfram Research (2020), JacobiEpsilon, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiEpsilon.html.
CMS
Wolfram Language. 2020. "JacobiEpsilon." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiEpsilon.html.
APA
Wolfram Language. (2020). JacobiEpsilon. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiEpsilon.html
BibTeX
@misc{reference.wolfram_2025_jacobiepsilon, author="Wolfram Research", title="{JacobiEpsilon}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiEpsilon.html}", note=[Accessed: 08-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_jacobiepsilon, organization={Wolfram Research}, title={JacobiEpsilon}, year={2020}, url={https://reference.wolfram.com/language/ref/JacobiEpsilon.html}, note=[Accessed: 08-January-2026]}