gives the complete elliptic integral .


gives the elliptic integral of the second kind .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For , .
  • .
  • EllipticE[m] has a branch cut discontinuity in the complex m plane running from to .
  • EllipticE[ϕ,m] has a branch cut discontinuity running along the ray from to infinity.
  • For certain special arguments, EllipticE automatically evaluates to exact values.
  • EllipticE can be evaluated to arbitrary numerical precision.
  • EllipticE automatically threads over lists.


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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  (33)

Numerical Evaluation  (4)

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:

EllipticE threads elementwise over lists:

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[{{x, +, {ⅈ,  , y}}}, EllipticE]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}}, EllipticE]:

Function Properties  (3)

EllipticE is defined for all real values less than or equal to 1:

EllipticE takes all real values greater than or equal to 1:

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

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:


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  (8)

Compute elliptic integrals:

Plot an incomplete elliptic integral over the complex plane:

Perimeter length of an ellipse:

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:

Properties & Relations  (4)

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  (2)

The defining integral converges only under additional conditions:

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 EllipticE:

Introduced in 1988
Updated in 1996