gives the complete elliptic integral .
gives the elliptic integral of the second kind .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For and , .
- 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. »
Examplesopen allclose all
Basic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (5)
Evaluate numerically for complex arguments:
The precision of the output tracks the precision of the input:
Evaluate EllipticE efficiently at high precision:
EllipticE threads elementwise over lists:
EllipticE can be used with Interval and CenteredInterval objects:
Specific Values (4)
Function Properties (10)
is defined for all real values less than or equal to 1:
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:
is nonincreasing on its domain:
is non-negative on its domain:
Indefinite integral of EllipticE:
Definite integral of an odd function over an interval centered at the origin is 0:
Series Expansions (4)
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:
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:
Properties & Relations (6)
EllipticE[ϕ,m] is real‐valued for real argument subject to the following conditions:
For real arguments, if , then 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 (2)
The defining integral converges only under additional conditions:
Different conventions exist for the second argument:
Wolfram Research (1988), EllipticE, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticE.html (updated 2022).
Wolfram Language. 1988. "EllipticE." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/EllipticE.html.
Wolfram Language. (1988). EllipticE. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EllipticE.html