Built-in Mathematica Symbol

EllipticE

EllipticE[m]
gives the complete elliptic integral

.

EllipticE[

, m]
gives the elliptic integral of the second kind

.

MORE INFORMATION

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[, 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.

Scope (14)

Evaluate numerically for complex arguments:

Evaluate to high precision:

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

Simple exact values are generated automatically:

EllipticE threads element-wise over lists:

Find series expansions at branch points:

Find limits at branch cuts:

TraditionalForm formatting:

Evaluate numerically for complex arguments:

Evaluate to high precision:

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

Simple exact values are generated automatically:

EllipticE threads element-wise over lists:

Series expansion:

Expand in series with respect to the modulus:

TraditionalForm formatting:

Generalizations & Extensions (1)

EllipticE can be applied to a power series:

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 (6)

Expand special cases:

Expand special cases under argument restrictions:

Numerically find a root of a transcendental equation:

Integrals:

Laplace transforms:

Limits on branch cuts:

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

Possible Issues (3)

The defining integral converges only under additional conditions:

Different conventions exist for the second argument:

In traditional form the vertical separator must be used:

Neat Examples (4)

Nested derivatives and integrals:

Plot EllipticE at integer points:

Calculate EllipticE through an analytically continued Taylor series:

Riemann surface of EllipticE: