scipy.special.hyp2f1 — SciPy v1.15.2 Manual (original) (raw)
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
confluent hypergeometric limit function.
Kummer’s (confluent hypergeometric) function.
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 z is described in [2], except for z in 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
[2]
- Zhang and J.M. Jin, “Computation of Special Functions”, Wiley 1996
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