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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 floating-point value is returned if both arguments are numerical and at least one of them is a floating-point 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.

Environment Interactions

When called with floating-point arguments, the function is sensitive to the environment variable DIGITS which determines the numerical working precision.


Example 1

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 floating-point arguments:

beta(3.5, sqrt(2)), beta(sqrt(2), 2.0 + 10.0*I)

Example 2

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)

Example 3

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))

Example 4

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)

Return Values

Arithmetical expression or a floating-point interval.

Overloaded By


See Also

MuPAD Functions

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