scipy.special.hyp2f1#
- scipy.special.hyp2f1(a, b, c, z, out=None) = <ufunc 'hyp2f1'>#
- Gauss hypergeometric function 2F1(a, b; c; z) - Parameters:
- a, b, carray_like
- Arguments, should be real-valued. 
- zarray_like
- Argument, real or complex. 
- outndarray, optional
- Optional output array for the function values 
 
- Returns:
- hyp2f1scalar or ndarray
- The values of the gaussian hypergeometric function. 
 
 - See also - Notes - This function is defined for \(|z| < 1\) as \[\mathrm{hyp2f1}(a, b, c, z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!},\]- and defined on the rest of the complex z-plane by analytic continuation [1]. Here \((\cdot)_n\) is the Pochhammer symbol; see - poch. When \(n\) is an integer the result is a polynomial of degree \(n\).- The implementation for complex values of - zis described in [2], except for- zin the region defined by\[0.9 <= \left|z\right| < 1.1, \left|1 - z\right| >= 0.9, \mathrm{real}(z) >= 0\]- in which the implementation follows [4]. - References [1]- NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/15.2 [2]- Zhang and J.M. Jin, “Computation of Special Functions”, Wiley 1996 
 [3]- Cephes Mathematical Functions Library, http://www.netlib.org/cephes/ [4]- J.L. Lopez and N.M. Temme, “New series expansions of the Gauss hypergeometric function”, Adv Comput Math 39, 349-365 (2013). https://doi.org/10.1007/s10444-012-9283-y - Examples - >>> import numpy as np >>> import scipy.special as sc - It has poles when c is a negative integer. - >>> sc.hyp2f1(1, 1, -2, 1) inf - It is a polynomial when a or b is a negative integer. - >>> a, b, c = -1, 1, 1.5 >>> z = np.linspace(0, 1, 5) >>> sc.hyp2f1(a, b, c, z) array([1. , 0.83333333, 0.66666667, 0.5 , 0.33333333]) >>> 1 + a * b * z / c array([1. , 0.83333333, 0.66666667, 0.5 , 0.33333333]) - It is symmetric in a and b. - >>> a = np.linspace(0, 1, 5) >>> b = np.linspace(0, 1, 5) >>> sc.hyp2f1(a, b, 1, 0.5) array([1. , 1.03997334, 1.1803406 , 1.47074441, 2. ]) >>> sc.hyp2f1(b, a, 1, 0.5) array([1. , 1.03997334, 1.1803406 , 1.47074441, 2. ]) - It contains many other functions as special cases. - >>> z = 0.5 >>> sc.hyp2f1(1, 1, 2, z) 1.3862943611198901 >>> -np.log(1 - z) / z 1.3862943611198906 - >>> sc.hyp2f1(0.5, 1, 1.5, z**2) 1.098612288668109 >>> np.log((1 + z) / (1 - z)) / (2 * z) 1.0986122886681098 - >>> sc.hyp2f1(0.5, 1, 1.5, -z**2) 0.9272952180016117 >>> np.arctan(z) / z 0.9272952180016122