# EllipticF

EllipticF[ϕ,m]

gives the elliptic integral of the first kind .

# Examples

open allclose all

## 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(36)

### Numerical Evaluation(5)

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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix EllipticF function using MatrixFunction:

### 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 :

### Function Properties(10)

is defined for all real values:

takes all real values:

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

is an analytic function of :

Has no singularities or discontinuities:

is not a meromorphic function of and :

is nondecreasing on its real domain:

is injective:

is not surjective:

is neither non-negative nor non-positive:

is neither convex nor concave:

### 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:

Wolfram Research (1988), EllipticF, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticF.html (updated 2022).

#### Text

Wolfram Research (1988), EllipticF, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticF.html (updated 2022).

#### CMS

Wolfram Language. 1988. "EllipticF." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/EllipticF.html.

#### APA

Wolfram Language. (1988). EllipticF. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EllipticF.html

#### BibTeX

@misc{reference.wolfram_2024_ellipticf, author="Wolfram Research", title="{EllipticF}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticF.html}", note=[Accessed: 16-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_ellipticf, organization={Wolfram Research}, title={EllipticF}, year={2022}, url={https://reference.wolfram.com/language/ref/EllipticF.html}, note=[Accessed: 16-September-2024 ]}