ellippy.carlson.elliprc

ellippy.carlson.elliprc(x, y)

Computes Carlson degenerate integral RC.

\[R_C(x, y) = \tfrac{1}{2} \int_0^{\infty} \frac{\mathrm{d}t}{(t+y)\,\sqrt{t+x}}\]

Parameters:

• x (ArrayLike) – Real-valued parameter.

• y (ArrayLike) – Real-valued parameter.

Returns:

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

Raises:

ValueError – If x < 0 or y = 0, or inputs contain NaN.

Graph

Special Cases

• RC(x, x) = 1/sqrt(x)

• RC(0, y) = π/(2*sqrt(y))

• RC(x, y) = atan(sqrt(y-x)/x) / sqrt(y-x) for y > x

• RC(x, y) = ln(sqrt(x) + sqrt(x-y)) / sqrt(x-y) for y < x

• RC(x, y) = 0 for x = ∞ or y = ∞

Related Functions

• RC(x, y) = RF(x, y, y)

Notes

RC is a degenerate case of the RF. It is an elementary function rather than an elliptic integral.

References

• Maddock, John, Paul Bristow, Hubert Holin, and Xiaogang Zhang. “Boost Math Library: Special Functions - Elliptic Integrals.” Accessed April 17, 2025. https://www.boost.org/doc/libs/1_88_0/libs/math/doc/html/math_toolkit/ellint.html.

• Carlson, B. C. “DLMF: Chapter 19 Elliptic Integrals.” Accessed February 19, 2025. https://dlmf.nist.gov/19.

• The SciPy Community. “SciPy: Special Functions - Elliprc.” Accessed April 17, 2025. https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.elliprc.html.

• Pornsiriprasert, Sira. Ellip: Elliptic Integrals for Rust. V. 0.5.1. Released October 10, 2025. https://docs.rs/ellip/0.5.1/ellip/index.html.