gives the elliptic integral of the first kind .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For real , and , TemplateBox[{phi, m}, EllipticF]=int_0^phi(1-m sin^2(theta))^(-1/2)dtheta.
  • 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.


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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:

EllipticF threads elementwise over lists:

Specific Values  (5)

Simple exact values are generated automatically:

Value at infinity:

Limiting values on branch cuts:

Find the root of the equation TemplateBox[{pi, m}, EllipticF]=3:

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 TemplateBox[{{x, +, {ⅈ,  , y}}, 1}, EllipticF]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}, 1}, EllipticF]:

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 TemplateBox[{phi, 1}, EllipticF] around :

Expand in series with respect to the modulus:

Plots of the first three approximations for TemplateBox[{pi, m}, EllipticF] 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:

TraditionalForm formatting:

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 phi=TemplateBox[{u, m}, JacobiAmplitude], :

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 2020).


Wolfram Research (1988), EllipticF, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticF.html (updated 2020).


@misc{reference.wolfram_2021_ellipticf, author="Wolfram Research", title="{EllipticF}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/EllipticF.html}", note=[Accessed: 17-September-2021 ]}


@online{reference.wolfram_2021_ellipticf, organization={Wolfram Research}, title={EllipticF}, year={2020}, url={https://reference.wolfram.com/language/ref/EllipticF.html}, note=[Accessed: 17-September-2021 ]}


Wolfram Language. 1988. "EllipticF." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/EllipticF.html.


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