beta -- the beta function
Introductionbeta(x, y) represents the beta function
gamma(x)*gamma(y)/gamma(x+y).
Call(s)beta(x, y)
Parametersx, y |
- | arithmetical expressions |
Returnsan arithmetical expression.
x
Side
EffectsWhen called with floating point arguments, the function is sensitive
to the environment variable DIGITS which determines the numerical
working precision.
Related
Functions
DetailsType::Numeric. Note that the beta
function may have a regular value, even if gamma(x) or
gamma(y) and gamma(x+y) are singular. In such
cases beta returns the limit of the quotients of the
singular terms.beta is returned, if none of
the arguments vanishes and at least one of the arguments does not
evaluate to a number of type Type::Numeric.
Example
1We demonstrate some calls with exact and symbolic input data:
>> beta(1, 5), beta(I, 3/2), beta(1, y + 1), beta(x, y)
1/2
PI gamma(I) 1
1/5, ----------------, -----, beta(x, y)
2 gamma(3/2 + I) y + 1
Floating point values are computed for floating point arguments:
>> beta(3.5, sqrt(2)), beta(sqrt(2), 2.0 + 10.0*I)
0.1395855454, - 0.01112350756 - 0.03108193098 I
Example
2The gamma function is singular if its argument is a
nonpositive integer. Nevertheless, beta has a regular
value for the following arguments:
>> beta(-3, 2)
1/6
Example
3The functions diff, expand and float handle expressions involving
beta:
>> diff(beta(x^2, x), x)
2 2 2
beta(x, x ) (psi(x) + 2 x psi(x ) - psi(x + x ) (2 x + 1))
>> expand(beta(x - 1, y + 1))
y gamma(x) gamma(y)
--------------------
gamma(x + y) (x - 1)
>> float(beta(100, 1000))
7.730325902e-147