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EllipticE [m]

gives the complete elliptic integral TemplateBox[{m}, EllipticE].

EllipticE [ϕ,m]

gives the elliptic integral of the second kind TemplateBox[{phi, m}, EllipticE2].

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Show More Show More
Function Properties  
Differentiation  
Integration  
Series Expansions  
Integral Transforms  
Function Representations  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
Related Guides
Related Links
History
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EllipticE [m]

gives the complete elliptic integral TemplateBox[{m}, EllipticE].

EllipticE [ϕ,m]

gives the elliptic integral of the second kind TemplateBox[{phi, m}, EllipticE2].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For and , TemplateBox[{phi, m}, EllipticE2]=int_0^phi(1-m sin^2(theta))^(1/2)dtheta.
  • TemplateBox[{m}, EllipticE]=TemplateBox[{{pi, /, 2}, m}, EllipticE2].
  • EllipticE [m] has a branch cut discontinuity in the complex m plane running from to .
  • EllipticE [ϕ,m] has branch cut discontinuities at and at .
  • For certain special arguments, EllipticE automatically evaluates to exact values.
  • EllipticE can be evaluated to arbitrary numerical precision.
  • EllipticE automatically threads over lists.
  • EllipticE can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity :

Scope  (41)

Numerical Evaluation  (5)

Evaluate numerically for complex arguments:

Evaluate to high precision:

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

Evaluate EllipticE efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around :

Compute the elementwise values of an array:

Or compute the matrix EllipticE function using MatrixFunction :

Specific Values  (4)

Simple exact values are generated automatically:

Find limiting values at branch cuts of the complete elliptic integral:

Find limiting values at branch cuts of the elliptic integral of the second kind:

Values at infinity:

Find the root of the equation TemplateBox[{m}, EllipticE]=2:

Visualization  (3)

Plot the complete elliptic integral:

Plot the elliptic integral of the second kind:

Plot the real part of TemplateBox[{z}, EllipticE]:

Plot the imaginary part of TemplateBox[{z}, EllipticE]:

Function Properties  (10)

TemplateBox[{m}, EllipticE] is defined for all real values less than or equal to 1:

TemplateBox[{m}, EllipticE] takes all real values greater than or equal to 1:

EllipticE is an odd function with respect to its first parameter:

EllipticE is not an analytic function:

Has both singularities and discontinuities for :

EllipticE is not a meromorphic function:

TemplateBox[{m}, EllipticE] is nonincreasing on its domain:

TemplateBox[{m}, EllipticE] is injective:

TemplateBox[{m}, EllipticE] is not surjective:

TemplateBox[{m}, EllipticE] is non-negative on its domain:

TemplateBox[{m}, EllipticE] is concave on its domain:

Differentiation  (4)

First derivative:

Higher derivatives:

Formula for the n^(th) derivative:

Derivative with respect to the first argument of the elliptic integral of the second kind:

Integration  (3)

Indefinite integral of EllipticE :

Definite integral of an odd function over an interval centered at the origin is 0:

More integrals:

Series Expansions  (4)

Taylor expansion for EllipticE :

Plot the first three approximations for EllipticE around :

Find series expansions at branch points:

Series expansion for the elliptic integral of the second kind:

Expand in series with respect to the modulus:

EllipticE can be applied to a power series:

Integral Transforms  (2)

Compute the Laplace transform using LaplaceTransform :

HankelTransform :

Function Representations  (6)

The definition of the elliptic integral of the second kind:

Complete elliptic integral is a partial case of the elliptic integral of the second kind:

Relation to other elliptic integrals:

Represent in terms of MeijerG using MeijerGReduce :

EllipticE can be represented as a DifferentialRoot :

TraditionalForm formatting:

Applications  (10)

Compute elliptic integrals:

Plot an incomplete elliptic integral over the complex plane:

Perimeter length of an ellipse:

Use ArcLength to obtain the perimeter:

Series expansion for almost equal axes lengths:

Compare with an approximation by Ramanujan:

Arc length of a hyperbola as a function of the angle of a point on the hyperbola:

Plot the arc length as a function of the angle:

Vector potential of a ring current in cylindrical coordinates:

The vertical and radial components of the magnetic field:

Plot magnitude of the magnetic field:

Inductance of a solenoid of radius r and length a with fixed numbers of turns per unit length:

Inductance per unit length of the infinite solenoid:

Calculate the surface area of a triaxial ellipsoid:

The area of an ellipsoid with half axes 3, 2, 1:

Calculate the area by integrating the differential surface elements:

Parametrization of a Mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):

Plot the resulting balloon:

Calculate the ratio of the main curvatures:

Express the radius of the original sheets in terms of the radius of the inflated balloon:

Parametrize an ellipse using EllipticE :

Plot using elliptic parametrization and circular parametrization:

Define the Halphen constant using elliptic integrals [MathWorld]:

Find the extended precision value:

Verify that it is zero of the function :

Properties & Relations  (6)

EllipticE [ϕ,m] is real‐valued for real argument subject to the following conditions:

For real arguments, if phi=TemplateBox[{u, m}, JacobiAmplitude], then TemplateBox[{u, m}, JacobiEpsilon]=TemplateBox[{phi, m}, EllipticE2] for :

For , this is only true for :

Expand special cases:

Expand special cases under argument restrictions:

Numerically find a root of a transcendental equation:

Limits on branch cuts:

EllipticE is automatically returned as a special case for some special functions:

Possible Issues  (1)

Different conventions exist for the second argument:

Neat Examples  (4)

Nested derivatives and integrals:

Plot EllipticE at integer points:

Calculate EllipticE through an analytically continued Taylor series:

Riemann surface of TemplateBox[{m}, EllipticE]:

History

Introduced in 1988 (1.0) | Updated in 1996 (3.0) ▪ 2020 (12.2) ▪ 2021 (13.0) ▪ 2022 (13.1)

Wolfram Research (1988), EllipticE, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticE.html (updated 2022).

Text

Wolfram Research (1988), EllipticE, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticE.html (updated 2022).

CMS

Wolfram Language. 1988. "EllipticE." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/EllipticE.html.

APA

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

BibTeX

@misc{reference.wolfram_2025_elliptice, author="Wolfram Research", title="{EllipticE}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticE.html}", note=[Accessed: 17-November-2025]}

BibLaTeX

@online{reference.wolfram_2025_elliptice, organization={Wolfram Research}, title={EllipticE}, year={2022}, url={https://reference.wolfram.com/language/ref/EllipticE.html}, note=[Accessed: 17-November-2025]}
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