# EllipticF EllipticF[ϕ,m]

gives the elliptic integral of the first kind .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• For real , and , .
• The complete elliptic integral associated with EllipticF is EllipticK.
• EllipticF is the inverse of JacobiAmplitude for real arguments. If , then for .
• EllipticF[ϕ,m] has branch discontinuity at and at .
• For certain special arguments, EllipticF automatically evaluates to exact values.
• EllipticF can be evaluated to arbitrary numerical precision.
• EllipticF automatically threads over lists.

# Examples

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## Basic Examples(4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

## Scope(25)

### Numerical Evaluation(4)

Evaluate for complex arguments:

Evaluate to high precision:

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

Evaluate EllipticF efficiently at high precision:

### Specific Values(5)

Simple exact values are generated automatically:

Value at infinity:

Limiting values on branch cuts:

Find the root of the equation :

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

### Visualization(3)

Plot the elliptic integral for various values of parameter :

Plot the elliptic integral as a function of its parameter :

Plot the real part of :

Plot the imaginary part of :

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Differentiate with respect to parameter :

### Integration(3)

Indefinite integral of EllipticF:

Definite integral of an odd function over an interval centered at the origin is 0:

More integrals:

### Series Expansions(3)

Taylor expansion for EllipticF:

Plots of the first three approximations for around :

Expand in series with respect to the modulus:

Plots of the first three approximations for around :

EllipticF can be applied to power series:

### Function Representations(4)

The definition of the elliptic integral of the second kind:

Relation to EllipticPi:

EllipticF can be represented as a DifferentialRoot:

## Applications(5)

Carry out an elliptic integral:

Plot an incomplete elliptic integral over the complex plane:

Calculate the surface area of a triaxial ellipsoid:

The area of an ellipsoid with half axes 3, 2, 1:

Calculate volume through integrating the differential surface elements:

Arc length parametrization of a curve that minimizes the integral of the square of its curvature:

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 through the radius of the inflated balloon:

## Properties & Relations(7)

EllipticF[ϕ,m] is real valued for real arguments subject to the following conditions:

Expand special cases:

Expand special cases under argument restrictions:

Compositions with the inverse function need PowerExpand:

Solve an equation containing EllipticF: Numerically find a root of a transcendental equation:

Limits on branch cuts:

For and , :

For , this is only true for :

## Possible Issues(2)

The defining integral converges only under additional conditions:

Different conventions exist for the second argument:

## Neat Examples(2)

Nested derivatives:

Plot EllipticF at integer points: