beta() [6/13]

template<typename T1 , typename T2 , require_all_arithmetic_t< T1, T2 > * = nullptr>

return_type_t< T1, T2 > stan::math::beta ( const T1  a, const T2  b  )

inline

Return the beta function applied to the specified arguments.

The beta function is defined for \(a > 0\) and \(b > 0\) by

\(\mbox{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\).

\[ \mbox{beta}(\alpha, \beta) = \begin{cases} \int_0^1 u^{\alpha - 1} (1 - u)^{\beta - 1} \, du & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \alpha} = \begin{cases} \left(\psi(\alpha)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{beta}(\alpha, \beta)}{\partial \beta} = \begin{cases} \left(\psi(\beta)-\psi(\alpha+\beta)\right)*\mbox{beta}(\alpha, \beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

Template Parameters

T1	type of first value
T2	type of second value

Parameters

a	First value
b	Second value

Returns

Beta function applied to the two values.

Definition at line 53 of file beta.hpp.