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Hypergeometric2F1

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Hypergeometric2F1[a, b, c, z]
    is the Gauss hypergeometric 2F1, equal to HypergeometricPFQ[{a, b}, {c}, z].

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Hypergeometric2F1[1, 1, 2, z]
Out[1]= -Log[1 - z]/z

Implementation notes

Attributes: NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Hypergeometric2F1[1, 1, 2, z]
Out[1]= -Log[1 - z]/z

A non-positive integer upper parameter terminates the Gauss series to a polynomial — here (1 - z)^3:

In[1]:= Hypergeometric2F1[-3, 1, 1, z]
Out[1]= 1 - 3 z + 3 z^2 - z^3

The function reproduces the elementary inverse trig functions: z * 2F1[1/2, 1/2, 3/2, z^2] = ArcSin[z]. Checked at z = 1/2 to 40 digits:

In[1]:= N[Hypergeometric2F1[1/2, 1/2, 3/2, 1/4]/2, 40]
Out[1]= 0.52359877559829887307710723054658381403285

In[2]:= N[ArcSin[1/2], 40]
Out[2]= 0.52359877559829887307710723054658381403285

Notes

Hypergeometric2F1[a, b, c, z] is the Gauss hypergeometric function HypergeometricPFQ[{a, b}, {c}, z], convergent for |z| < 1 (and by termination for non-positive integer a or b). Many elementary functions are special cases: 2F1[1, 1, 2, z] = -Log[1 - z]/z and z * 2F1[1/2, 1/2, 3/2, z^2] = ArcSin[z].