scipy.special.elliprj(x, y, z, p, out=None) = <ufunc 'elliprj'>#

Symmetric elliptic integral of the third kind.

The function RJ is defined as [1]

\[R_{\mathrm{J}}(x, y, z, p) = \frac{3}{2} \int_0^{+\infty} [(t + x) (t + y) (t + z)]^{-1/2} (t + p)^{-1} dt\]


This function should be considered experimental when the inputs are unbalanced. Check correctness with another independent implementation.

x, y, z, parray_like

Real or complex input parameters. x, y, or z are numbers in the complex plane cut along the negative real axis (subject to further constraints, see Notes), and at most one of them can be zero. p must be non-zero.

outndarray, optional

Optional output array for the function values

Rscalar or ndarray

Value of the integral. If all of x, y, z, and p are real, the return value is real. Otherwise, the return value is complex.

If p is real and negative, while x, y, and z are real, non-negative, and at most one of them is zero, the Cauchy principal value is returned. [1] [2]

See also


Degenerate symmetric integral.


Symmetric elliptic integral of the second kind.


Completely-symmetric elliptic integral of the first kind.


Completely-symmetric elliptic integral of the second kind.


The code implements Carlson’s algorithm based on the duplication theorems and series expansion up to the 7th order. [3] The algorithm is slightly different from its earlier incarnation as it appears in [1], in that the call to elliprc (or atan/atanh, see [4]) is no longer needed in the inner loop. Asymptotic approximations are used where arguments differ widely in the order of magnitude. [5]

The input values are subject to certain sufficient but not necessary constraints when input arguments are complex. Notably, x, y, and z must have non-negative real parts, unless two of them are non-negative and complex-conjugates to each other while the other is a real non-negative number. [1] If the inputs do not satisfy the sufficient condition described in Ref. [1] they are rejected outright with the output set to NaN.

In the case where one of x, y, and z is equal to p, the function elliprd should be preferred because of its less restrictive domain.

Added in version 1.8.0.


[1] (1,2,3,4,5)

B. C. Carlson, “Numerical computation of real or complex elliptic integrals,” Numer. Algorithm, vol. 10, no. 1, pp. 13-26, 1995.


B. C. Carlson, ed., Chapter 19 in “Digital Library of Mathematical Functions,” NIST, US Dept. of Commerce.


B. C. Carlson, J. FitzSimmons, “Reduction Theorems for Elliptic Integrands with the Square Root of Two Quadratic Factors,” J. Comput. Appl. Math., vol. 118, nos. 1-2, pp. 71-85, 2000.


F. Johansson, “Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms,” in J. Blumlein, C. Schneider, P. Paule, eds., “Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory,” pp. 269-293, 2019 (Cham, Switzerland: Springer Nature Switzerland)


B. C. Carlson, J. L. Gustafson, “Asymptotic Approximations for Symmetric Elliptic Integrals,” SIAM J. Math. Anls., vol. 25, no. 2, pp. 288-303, 1994.


Basic homogeneity property:

>>> import numpy as np
>>> from scipy.special import elliprj
>>> x = 1.2 + 3.4j
>>> y = 5.
>>> z = 6.
>>> p = 7.
>>> scale = 0.3 - 0.4j
>>> elliprj(scale*x, scale*y, scale*z, scale*p)
>>> elliprj(x, y, z, p)*np.power(scale, -1.5)

Reduction to simpler elliptic integral:

>>> elliprj(x, y, z, z)
>>> from scipy.special import elliprd
>>> elliprd(x, y, z)

All arguments coincide:

>>> elliprj(x, x, x, x)
>>> np.power(x, -1.5)