gives the elliptic integral of the first kind .
- 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.
Examplesopen allclose all
Basic Examples (4)
Numerical Evaluation (4)
Specific Values (5)
EllipticF is an odd function with respect to its first parameter:
Indefinite integral of EllipticF:
Series Expansions (3)
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:
Possible Issues (2)
The defining integral converges only under additional conditions:
Different conventions exist for the second argument:
Neat Examples (2)
Plot EllipticF at integer points:
Wolfram Research (1988), EllipticF, Wolfram Language function, https://reference.wolfram.com/language/ref/EllipticF.html (updated 2020).
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