# EllipticE

EllipticE[m]

gives the complete elliptic integral .

EllipticE[ϕ,m]

gives the elliptic integral of the second kind .

# Details • 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. »

# 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:

EllipticE can be used with Interval and CenteredInterval objects:

### 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 :

### Visualization(3)

Plot the complete elliptic integral:

Plot the elliptic integral of the second kind:

Plot the real part of :

Plot the imaginary part of :

### 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 injective: is not surjective: is non-negative on its domain: is concave on its domain:

### Differentiation(4)

First derivative:

Higher derivatives:

Formula for the 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:

EllipticE can be represented as a DifferentialRoot:

## Applications(8)

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 realvalued for real argument subject to the following conditions:

For real arguments, if , then 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(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: