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

  • EllipticE[ m ] gives the complete elliptic integral .
  • EllipticE[, m ] gives the elliptic integral of the second kind .
  • Mathematical function (see Section A.3.10).
  • For , .
  • .
  • See Section 3.2.11 for a discussion of argument conventions for elliptic integrals.
  • EllipticE[ m ] has a branch cut discontinuity in the complex m plane running from to .
  • EllipticE[, m ] has a branch cut discontinuity running along the ray from to infinity.
  • See the Mathematica book: Section 3.2.11.
  • See also: JacobiZeta, JacobiAmplitude.

    Further Examples

    This is the definition of the elliptic integral of the second 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 second kind, a particular case of the hypergeometric function.

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    This is the indefinite integral of the complete integral .

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    This is a series expansion around .

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    This is the derivative of the incomplete function with respect to .

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