Beta function
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beta(x
, y
)
beta(x, y)
represents the beta function
.
The beta function is defined for complex arguments x and y.
The result is expressed by calls to the gamma function if both
arguments are of type Type::Numeric
. Note that the beta function
may have a regular value, even if Γ(x) or Γ(y) and Γ(x + y) are
singular. In such cases beta
returns the limit
of the quotients of the singular terms.
A floatingpoint value is returned if both arguments are numerical and at least one of them is a floatingpoint value.
An unevaluated call of 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
.
When called with floatingpoint arguments, the function is sensitive
to the environment variable DIGITS
which determines the numerical
working precision.
We demonstrate some calls with exact and symbolic input data:
beta(1, 5), beta(I, 3/2), beta(1, y + 1), beta(x, y)
Floating point values are computed for floatingpoint arguments:
beta(3.5, sqrt(2)), beta(sqrt(2), 2.0 + 10.0*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)
The functions diff
, expand
and float
handle expressions
involving beta
:
diff(beta(x^2, x), x)
expand(beta(x  1, y + 1))
float(beta(100, 1000))
The functions diff
and series
can handle beta
:
diff(beta(x, y), x); diff(beta(x, y), y);
normal(series(beta(x, y), y = 0, 3))
series(beta(x, x), x = infinity, 4)

Arithmetical expression or a floatingpoint interval.
x