SciPy

scipy.special.hyp2f1

scipy.special.hyp2f1(a, b, c, z) = <ufunc 'hyp2f1'>

Gauss hypergeometric function 2F1(a, b; c; z)

Parameters:

a, b, c : array_like

Arguments, should be real-valued.

z : array_like

Argument, real or complex.

Returns:

hyp2f1 : scalar 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. 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 [R494].

References

[R494](1, 2)
  1. Zhang and J.M. Jin, “Computation of Special Functions”, Wiley 1996
[R495]Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html
[R496]NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/

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