This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# EllipticF

 EllipticFgives the elliptic integral of the first kind .
• Mathematical function, suitable for both symbolic and numerical manipulation.
• For , .
• EllipticF 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.
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 Scope   (7)
Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Simple exact values are generated automatically:
Expand in series with respect to the modulus:
EllipticF can be applied to power series:
 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:
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:
Integrals:
Limits on branch cuts:
The defining integral converges only under additional conditions:
Different conventions exist for the second argument:
In traditional form the vertical separator must be used:
Nested derivatives:
Plot EllipticF at integer points: