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

hyp0f1: confluent hypergeometric limit function.
hyp1f1: 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

[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