scipy.special.hyp2f1#

scipy.special.hyp2f1(a, b, c, z, out=None) = <ufunc 'hyp2f1'>#

Gauss hypergeometric function \({}_2F_1(a, b; c; z)\).

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

\[{}_2F_1(a, b; c; z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!},\]

and defined on the rest of the complex z-plane by analytic continuation [1]. Here \((\cdot)_n\) is the Pochhammer symbol; see poch. When \(a\) or \(b\) is a nonpositive integer, the series terminates and the result is a polynomial.

The implementation for complex values of z is described in [2], except for z in the region defined by

\[0.9 \le \left|z\right| < 1.1, \left|1 - z\right| \ge 0.9, \mathrm{Re}(z) \ge 0\]

in which the implementation follows [4].

Array API Standard Support

hyp2f1 has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	✅
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. hyp2f1 does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

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