ellippy.carlson.elliprg#

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

Computes Carlson symmetric integral RG.

\[R_G(x,y,z) = \frac{1}{4}\int_{0}^{\infty}\frac{t} {\sqrt{(t+x)(t+y)(t+z)}}\,\left(\dfrac{x}{t+x}+\dfrac{y}{t+y}+\dfrac{z}{t+z}\right)\,\mathrm{d}t\]
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 infinite, or inputs contain NaN.

Graph

Special Cases

  • RG(x, x, x) = sqrt(x)

  • RG(0, y, y) = π/4 * sqrt(y)

  • RG(x, y, y) = (y * RC(x, y) + sqrt(x))/2

  • RG(0, 0, z) = sqrt(z)/2

Related Functions

With c = csc²φ, r = 1/x², and kc² = 1 - m,

  • E(m) = 2·RG(0, kc², 1)

  • 5(φ, m) = 2·RG(c - 1, c - m, c) - (c - 1)·RF(c - 1, c - m, c) - sqrt((c - 1) * (c - m) / c)

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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