Gamma (FSharp.Stats)

Gamma Module

Namespace: FSharp.Stats.SpecialFunctions
Assembly: FSharp.Stats.dll

 Approximations for the gamma function and related functions.

 The gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension 
 of the factorial function to complex numbers:

 Γ(x) = (x-1)!

The gamma function is defined for all complex numbers except the non-positive integers.

Functions and values

Function or value	Description
`_gamma z` Full Usage: `_gamma z` Parameters: z : `^T` - The function input for approximating Γ(z) Returns: `^T` Modifiers: inline Type parameters: ^T	Computes an approximation of the real value of the gamma function using the Lanczos Coefficients described in Numerical Recipes (Press et al) The caller is responsible to handle edge cases such as nan, infinity, and -infinity in the input z : `^T` The function input for approximating Γ(z) Returns: `^T`
`_gammaLn z` Full Usage: `_gammaLn z` Parameters: z : `^T` - The function input for approximating ln(Γ(z)) Returns: `^T` Modifiers: inline Type parameters: ^T	Computes an approximation of the real value of the log gamma function using the Lanczos Coefficients described in Numerical Recipes (Press et al) The caller is responsible to handle edge cases such as nan, infinity, and -infinity in the input z : `^T` The function input for approximating ln(Γ(z)) Returns: `^T`
`digamma x` Full Usage: `digamma x` Parameters: x : `^T` Returns: `^T` Modifiers: inline Type parameters: ^T	Full digamma function ψ(x), x : `^T` Returns: `^T`
`digammaPositive x` Full Usage: `digammaPositive x` Parameters: x : `^T` - The input value (must be > 1). Returns: `^T` The value of the digamma function ψ(x). Modifiers: inline Type parameters: ^T	Optimized digamma function for x > 1. Skips reflection and tan-based correction. Use when x is guaranteed to be greater than 1. x : `^T` The input value (must be > 1). Returns: `^T` The value of the digamma function ψ(x).
`gamma z` Full Usage: `gamma z` Parameters: z : `'T` - The function input for approximating Γ(z) Returns: `'T` Modifiers: inline Type parameters: 'T	Computes an approximation of the real value of the gamma function using the Lanczos Coefficients described in Numerical Recipes (Press et al) Edge cases in the input (nan, infinity, and -infinity) are catched and handled. This might be slower than the unchecked version `_gamma` but does not require input sanitation to get expected results for these cases. z : `'T` The function input for approximating Γ(z) Returns: `'T`
`gammaLn z` Full Usage: `gammaLn z` Parameters: z : `'T` - The function input for approximating ln(Γ(z)) Returns: `'T` Modifiers: inline Type parameters: 'T	Computes an approximation of the real value of the log gamma function using the Lanczos Coefficients described in Numerical Recipes (Press et al) Edge cases in the input (nan, infinity, and -infinity) are catched and handled. This might be slower than the unchecked version `_gamma` but does not require input sanitation to get expected results for these cases. z : `'T` The function input for approximating ln(Γ(z)) Returns: `'T`
`gammpapprox a x psig` Full Usage: `gammpapprox a x psig` Parameters: a : `^T` x : `^T` psig : `bool` Returns: `^T` Modifiers: inline Type parameters: ^T	a : `^T` x : `^T` psig : `bool` Returns: `^T`
`gcf a x` Full Usage: `gcf a x` Parameters: a : `^T` x : `^T` Returns: `^T` Modifiers: inline Type parameters: ^T	a : `^T` x : `^T` Returns: `^T`
`gser a x` Full Usage: `gser a x` Parameters: a : `^T` x : `^T` Returns: `^T` Modifiers: inline Type parameters: ^T	a : `^T` x : `^T` Returns: `^T`
`lowerIncomplete a x` Full Usage: `lowerIncomplete a x` Parameters: a : `'T` - x : `'T` - Returns: `'T` Modifiers: inline Type parameters: 'T	Returns the lower incomplete gamma function gamma(a,x) = int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0. a : `'T` x : `'T` Returns: `'T` Example
`lowerIncompleteRegularized a x` Full Usage: `lowerIncompleteRegularized a x` Parameters: a : `'T` x : `'T` Returns: `'T` Modifiers: inline Type parameters: 'T	Returns the incomplete gamma function P(a,X) (regularized gamma) a : `'T` x : `'T` Returns: `'T`
`maximum` Full Usage: `maximum` Returns: `float`	Maximum gamma Returns: `float`
`trigamma x` Full Usage: `trigamma x` Parameters: x : `float` Returns: `float`	Trigamma function. This code has been adapted from the FORTRAN77 and subsequent C code by B. E. Schneider and John Burkardt. The code had been made public under the GNU LGPL license. x : `float` Returns: `float`
`upperIncomplete a x` Full Usage: `upperIncomplete a x` Parameters: a : `'T` - x : `'T` - Returns: `'T` Modifiers: inline Type parameters: 'T	Returns the upper incomplete gamma function Gamma(a,x) = int(exp(-t)t^(a-1),t=0..x) for real a > 0, x > 0. a : `'T` x : `'T` Returns: `'T` Example
`upperIncompleteRegularized a x` Full Usage: `upperIncompleteRegularized a x` Parameters: a : `'T` x : `'T` Returns: `'T` Modifiers: inline Type parameters: 'T	Returns the incomplete gamma function Q(a,X) = 1 - P(a,X) (regularized gamma) a : `'T` x : `'T` Returns: `'T`

Gamma Module

Functions and values

Example

Example