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 Documentation / Mathematica / Built-in Functions / Mathematical Functions / Elliptic Integrals  /
EllipticF

  • EllipticF[, m ] gives the elliptic integral of the first kind .
  • Mathematical function (see Section A.3.10).
  • For , .
  • The complete elliptic integral associated with EllipticF is EllipticK.
  • EllipticF is the inverse of JacobiAmplitude. If then .
  • EllipticF[, m ] has a branch cut discontinuity running along the ray from to infinity.
  • See Section 3.2.11 for a discussion of argument conventions for elliptic integrals.
  • See the Mathematica book: Section 3.2.11.
  • See also: JacobiZeta, JacobiAmplitude.

    Further Examples

    This is the definition of the elliptic integral of the first kind. Classically, the integral is defined for on the interval , but the function also makes sense for more general (complex-valued) .

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    Setting gives the complete elliptic integral of the first kind, which is traditionally denoted by a different letter, .

    In[2]:=

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    These are the partial derivatives.

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    In[4]:=

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