ellippy.carlson.elliprd#

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

Computes Carlson degenerate integral RD.

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

  • x (ArrayLike) – Real-valued parameter.

  • y (ArrayLike) – Real-valued parameter.

  • z (ArrayLike) – Real-valued parameter.

Returns:

Scalar or numpy.ndarray broadcast from inputs.

Raises:

ValueError – If x < 0, y < 0, z ≤ 0 or when both x and y are zero, or inputs contain NaN.

Graph

Special Cases

  • RD(x, x, x) = 1/(x sqrt(x))

  • RD(0, y, y) = 3/4 * π / (y sqrt(y))

  • RD(x, y, y) = 3/(2(y-x)) * (RC(x, y) - sqrt(x)/y) for x ≠ y

  • RD(x, x, z) = 3/(z-x) * (RC(z, x) - 1/sqrt(z)) for x ≠ z

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

Related Functions

  • With c = csc²φ: D(φ, m) = RD(c - 1, c - m, c) / 3

Notes

The parameters x and y (but not 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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