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Beta

Status: Stable

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

Description

Beta[a, b]
    is the Euler beta function B(a, b) = Gamma(a) Gamma(b) / Gamma(a+b).
Beta[z, a, b]
    is the incomplete beta function Integral_0^z t^(a-1) (1-t)^(b-1) dt.
Beta[z0, z1, a, b]
    is the generalized incomplete beta Beta[z1, a, b] - Beta[z0, a, b].
Exact for rational arguments (a positive integer gives a rational via
Pochhammer); non-positive integer poles give ComplexInfinity. Machine
and arbitrary-precision (MPFR) real and complex inputs evaluate
numerically. The incomplete form reduces through Hypergeometric2F1.
Listable.

Examples

No verified examples yet for this function.

Implementation notes

Attributes: Listable, NumericFunction, Protected, ReadProtected.

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]:= Beta[3, 5]
Out[1]= 1/105

The central value is exactly Pi, and rational orders fold into Gamma quotients:

In[1]:= Beta[1/2, 1/2]
Out[1]= Pi

In[2]:= Beta[1/3, 1/3]
Out[2]= Gamma[1/3]^2/Gamma[2/3]

Positive-integer orders give the reciprocal binomial relation 1/B(7, 3) = 9 C(8, 2):

In[1]:= Beta[7, 3]
Out[1]= 1/252

In[2]:= 1/Beta[7, 3] - 9 Binomial[8, 2]
Out[2]= 0

The three-argument incomplete beta, and arbitrary-precision numerics:

In[1]:= Beta[5, 2, 3]
Out[1]= 1025/12

In[2]:= N[Beta[2.5, 3.5], 30]
Out[2]= 0.036815538909255388078101134397

Notes

Beta[a, b] = Gamma[a] Gamma[b]/Gamma[a+b] is the Euler beta function. Beta[z, a, b] is the incomplete beta integral, and Beta[z0, z1, a, b] the generalized incomplete form. Exact for rational arguments via Pochhammer; non-positive integer poles give ComplexInfinity; the incomplete form reduces through Hypergeometric2F1. Real and complex inputs evaluate at machine or MPFR precision. Listable.