ellippy.carlson.elliprj#

ellippy.carlson.elliprj(x, y, z, p)[source]#

Computes Carlson symmetric integral RJ.

\[R_J(x, y, z, p) = \tfrac{3}{2} \int_0^{\infty} \frac{\mathrm{d}t}{(t+p)\,\sqrt{(t+x)(t+y)(t+z)}}\]
Parameters:

  • x (ArrayLike) – Real-valued parameter.

  • y (ArrayLike) – Real-valued parameter.

  • z (ArrayLike) – Real-valued parameter.

  • p (ArrayLike) – Real-valued parameter. p ∈ ℝ, p ≠ 0.

Returns:

Scalar or numpy.ndarray broadcast from inputs. Returns the Cauchy principal value if p < 0.

Raises:

ValueError – If any of x, y, or z is negative, or more than one of them are zero, or p = 0, or inputs contain NaN.

Graph

Special Cases

  • RJ(x, x, x, x) = 1/(x sqrt(x))

  • RJ(x, y, z, z) = RD(x, y, z)

  • RJ(x, x, x, p) = 3/(x-p) * (RC(x, p) - 1/sqrt(x)) for x ≠ p and xp ≠ 0

  • RJ(x, y, y, y) = RD(x, y, y)

  • RJ(x, y, y, p) = 3/(p-y) * (RC(x, y) - RC(x, p)) for y ≠ p

  • RJ(x, y, z, p) = 0 for x = ∞ or y = ∞ or z = ∞ or p = ∞

Related Functions

  • With c = csc²φ and kc² = 1 - m: Π(φ, n, m) = n / 3 * RJ(c - 1, c - m, c, c - n) + F(φ, m)

Notes

The parameters x, y, and z are symmetric. This means swapping them does not change the value of the function. At most one of them can be zero.

References

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