gives the elliptic integral of the first kind .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For , .
- The complete elliptic integral associated with EllipticF is EllipticK.
- EllipticF is the inverse of JacobiAmplitude. If then .
- EllipticF[ϕ,m] has a branch cut discontinuity running along the ray from to infinity.
- 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 (5)
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: