The Carlson elliptic integrals, also known as the Carlson symmetric forms, are a standard set of canonical elliptic integrals which provide a convenient alternative to Legendre's elliptic integrals of the first, second, and third kind. Carlson and Legendre elliptic integrals may be converted to each other.
The Carlson elliptic integrals are defined as
(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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They are implemented in the Wolfram Language as CarlsonRC[x, y], CarlsonRD[x, y, z], CarlsonRE[x, y], CarlsonRF[x, y, z], CarlsonRG[x, y, z], CarlsonRJ[x, y, z, rho], CarlsonRK[x, y], and CarlsonRM[x, y, rho].
For and , the incomplete elliptic integrals of the first, second, and third kind are related to the Carlson elliptic integrals by
(11)
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(12)
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(13)
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Expressing the complete Legendre-Jacobi integrals in terms of the incomplete Carlson integrals by plugging into the above gives
(14)
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(15)
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(Press and Teukolsky 1990) and
(17)
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(19)
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The functions also satisfy the following homogeneity properties:
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(21)
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(Press and Teukolsky 1990).
Special values include
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(23)
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(24)
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where is the lemniscate constant.