\(\text{B}\)

You can use the \(\text{B}\) function to calculate the beta function of a value.

You can use the \betafn backslash command to insert this function.

The following variants of this function are available:

• \(\text{complex } \text{B} \left ( \text{<s>}, \text{<z>} \right )\)

Note that the function is only defined for \(s > 0\) and \(z > 0\). The \(\text{B}\) function will return NaN for zero or negative values of \(s\) or \(z\).

For small positive real values of \(s\) and \(z\), this function calculates the beta function using the relation:

\[\text{B} \left ( s, z \right ) = \frac{\Gamma \left ( s \right ) \Gamma \left ( z \right )} {\Gamma \left ( s + z \right )}\]

For larger values of \(s\) and \(z\) or in cases where either \(s\) or \(z\) are complex, this function calculates values using the \(\text{lnGamma}\) function:

\[\text{B} \left ( s, z \right ) = \mathit{e} ^ { \left ( \text{ln } \Gamma \left ( s \right ) + \text{ln } \Gamma \left ( z \right ) - \text{ln } \Gamma \left ( s + z \right ) \right ) }\]

For real values of \(s\) and \(z\), the threshold used to determine how the \(\text{B}\) function is calculated is based on the magnitude of the expected value returned by the \(\Gamma\) function.

Note

The function name uses Greek upper case ‘’beta’’ as the function name, not the letter ‘’B’’.

Below is a simple example using the beta function:

\[a = \text{B} \left ( 2, 1 \right )\]

\[\text{a = 0.5}\]