maxframe.tensor.special.elliprg#
- maxframe.tensor.special.elliprg(x, y, z, **kwargs)[source]#
Completely-symmetric elliptic integral of the second kind.
The function RG is defined as [1]
\[R_{\mathrm{G}}(x, y, z) = \frac{1}{4} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} \left(\frac{x}{t + x} + \frac{y}{t + y} + \frac{z}{t + z}\right) t dt\]- Parameters:
x (array_like) – Real or complex input parameters. x, y, or z can be any number in the complex plane cut along the negative real axis.
y (array_like) – Real or complex input parameters. x, y, or z can be any number in the complex plane cut along the negative real axis.
z (array_like) – Real or complex input parameters. x, y, or z can be any number in the complex plane cut along the negative real axis.
out (ndarray, optional) – Optional output array for the function values
- Returns:
R – Value of the integral. If all of x, y, and z are real, the return value is real. Otherwise, the return value is complex.
- Return type:
scalar or ndarray
See also
Notes
The implementation uses the relation [1]
\[2 R_{\mathrm{G}}(x, y, z) = z R_{\mathrm{F}}(x, y, z) - \frac{1}{3} (x - z) (y - z) R_{\mathrm{D}}(x, y, z) + \sqrt{\frac{x y}{z}}\]and the symmetry of x, y, z when at least one non-zero parameter can be chosen as the pivot. When one of the arguments is close to zero, the AGM method is applied instead. Other special cases are computed following Ref. [2]
References