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factorialln

Natural logarithm of the factorial function.

The natural logarithm of the factorial function may be expressed

$$f(n)=\ln (n!)$$

The factorial function may be defined as the product

$$n! = \prod_{k=1}^n k$$

or according to the recurrence relation

$$n! = \begin{cases}1 & \textrm{if } n = 0,\\(n-1)! \times n & \textrm{if } n > 1\end{cases}$$

Following the convention for an empty product, in all definitions,

$$0! = 1$$

The Gamma function extends the factorial function for non-integer values.

$$n! = \Gamma(n+1)$$

The factorial of a negative integer is not defined.

Evaluating the natural logarithm of factorial function is useful as the factorial function can overflow for large n. Thus, factorialln( n ) is generally preferred to ln( n! ).

Installation

npm install @stdlib/math-base-special-factorialln

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var factorialln = require( '@stdlib/math-base-special-factorialln' );

factorialln( x )

Evaluates the natural logarithm of the factorial function. For input values other than negative integers, the function returns ln( x! ) = ln( Γ(x+1) ), where Γ is the Gamma function. For negative integers, the function returns NaN.

var v = factorialln( 3.0 );
// returns ~1.792

v = factorialln( 2.4 );
// returns ~1.092

v = factorialln( -1.0 );
// returns NaN

v = factorialln( -1.5 );
// returns ~1.266

If provided NaN, the function returns NaN.

var v = factorialln( NaN );
// returns NaN

Examples

var incrspace = require( '@stdlib/array-base-incrspace' );
var factorialln = require( '@stdlib/math-base-special-factorialln' );

var x = incrspace( -10.0, 50.0, 0.5 );

var i;
for ( i = 0; i < x.length; i++ ) {
    console.log( 'x: %d, f(x): %d', x[ i ], factorialln( x[ i ] ) );
}

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C APIs

Usage

#include "stdlib/math/base/special/factorialln.h"

stdlib_base_factorialln( x )

Evaluates the natural logarithm of the factorial function. For input values other than negative integers, the function returns ln( x! ) = ln( Γ(x+1) ), where Γ is the Gamma function. For negative integers, the function returns NaN.

double out = stdlib_base_factorialln( 3.0 );
// returns ~1.792

out = stdlib_base_factorialln( -1.5 );
// returns ~1.266

The function accepts the following arguments:

• x: [in] double input value.

double stdlib_base_factorialln( const double n );

Examples

#include "stdlib/math/base/special/factorialln.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { 2.0, 3.0, 5.0, 8.0 };

    double y;
    int i;
    for ( i = 0; i < 4; i++ ) {
        y = stdlib_base_factorialln( x[ i ] );
        printf( "factorialln(%lf) = %lf\n", x[ i ], y );
    }
}

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See Also

• @stdlib/math-base/special/factorial: evaluate a factorial.

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.