beta
-- the beta functionbeta
(x, y)
represents the beta function
gamma(x)*gamma(y)/gamma(x+y).
beta(x, y)
x, y |
- | arithmetical expressions |
an arithmetical expression.
x
When called with floating point arguments, the function is sensitive
to the environment variable DIGITS
which determines the numerical
working precision.
Type::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
.We 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
The 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
The 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