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.
Examplesopen allclose all
Basic Examples (5)
Series expansion at Infinity:
Numerical Evaluation (4)
Specific Values (4)
Function Properties (3)
Indefinite integral of EllipticE:
Series Expansions (4)
Function Representations (6)
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:
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:
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 2020).
Wolfram Language. 1988. "EllipticE." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. 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