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 analyticcontinuation[1].Here\((\cdot)_n\) is the Pochhammer symbol; seepoch. When\(n\) is an integer the result is a polynomial of degree\(n\).

The implementation for complex values ofz is described in[2],except forz 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 Functionshttps://dlmf.nist.gov/15.2

[2]

  1. 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 Gausshypergeometric function”, Adv Comput Math 39, 349-365 (2013).https://doi.org/10.1007/s10444-012-9283-y

Examples

>>>importnumpyasnp>>>importscipy.specialassc

It has poles whenc is a negative integer.

>>>sc.hyp2f1(1,1,-2,1)inf

It is a polynomial whena orb 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/carray([1.        , 0.83333333, 0.66666667, 0.5       , 0.33333333])

It is symmetric ina andb.

>>>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)/z1.3862943611198906
>>>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)/z0.9272952180016122
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