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`beta`

Beta function

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Syntax

```beta(`x`, `y`)
```

Description

`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.

Examples

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)`

Parameters

 `x`, `y`

Return Values

Arithmetical expression or a floating-point interval.

`x`