ellippy.carlson.elliprf#

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

Computes Carlson symmetric integral RF.

\[R_F(x, y, z) = \tfrac{1}{2} \int_0^{\infty} \frac{\mathrm{d}t}{\sqrt{(t+x)(t+y)(t+z)}}\]
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 any of x, y, or z is negative, or more than one of them are zero, or inputs contain NaN.

Graph

Special Cases

  • RF(x, x, x) = 1/sqrt(x)

  • RF(x, y, y) = RC(x, y)

  • RF(0, y, y) = π/(2 sqrt(y))

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

Related Functions

With c = csc²φ, r = 1/x², and kc² = 1 - m, - F(φ,m) = RF(c - 1, c - m, c) - el1(x, kc) = RF(r, r + m, r + 1)

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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