scipy.special.beta

scipy.special.beta(a, b, out=None) = <ufunc 'beta'>

Beta function.

This function is defined in [1] as

\[B(a, b) = \int_0^1 t^{a-1}(1-t)^{b-1}dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},\]

where \(\Gamma\) is the gamma function.

Parameters

a, barray-like

Real-valued arguments

outndarray, optional

Optional output array for the function result

Returns

scalar or ndarray

Value of the beta function

See also

gamma

the gamma function

betainc

the incomplete beta function

betaln

the natural logarithm of the absolute value of the beta function

References

1

NIST Digital Library of Mathematical Functions, Eq. 5.12.1. https://dlmf.nist.gov/5.12

Examples

>>> import scipy.special as sc

The beta function relates to the gamma function by the definition given above:

>>> sc.beta(2, 3)
0.08333333333333333
>>> sc.gamma(2)*sc.gamma(3)/sc.gamma(2 + 3)
0.08333333333333333

As this relationship demonstrates, the beta function is symmetric:

>>> sc.beta(1.7, 2.4)
0.16567527689031739
>>> sc.beta(2.4, 1.7)
0.16567527689031739

This function satisfies \(B(1, b) = 1/b\):

>>> sc.beta(1, 4)
0.25

scipy.special.gammainccinv scipy.special.betaln