time-to-botec

README.md (5750B)

---

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 # factorialln
 
 > Natural logarithm of the [factorial function][factorial-function].
 
 <section class="intro">
 
 The natural logarithm of the factorial function may be expressed
 
 <!-- <equation class="equation" label="eq:factorialln_function" align="center" raw="f(n)=\ln (n!)" alt="Equation of the natural logarithm of the factorial."> -->
 
 <div class="equation" align="center" data-raw-text="f(n)=\ln (n!)" data-equation="eq:factorialln_function">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_factorialln_function.svg" alt="Equation of the natural logarithm of the factorial.">
     <br>
 </div>
 
 <!-- </equation> -->
 
 The [factorial function][factorial-function] may be defined as the product
 
 <!-- <equation class="equation" label="eq:factorial_function" align="center" raw="n! = \prod_{k=1}^n k" alt="Factorial function definition"> -->
 
 <div class="equation" align="center" data-raw-text="n! = \prod_{k=1}^n k" data-equation="eq:factorial_function">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_factorial_function.svg" alt="Factorial function definition">
     <br>
 </div>
 
 <!-- </equation> -->
 
 or according to the recurrence relation
 
 <!-- <equation class="equation" label="eq:factorial_recurrence_relation" align="center" raw="n! = \begin{cases}1 & \textrm{if } n = 0,\\(n-1)! \times n & \textrm{if } n > 1\end{cases}" alt="Factorial function recurrence relation"> -->
 
 <div class="equation" align="center" data-raw-text="n! = \begin{cases}1 &amp; \textrm{if } n = 0,\\(n-1)! \times n &amp; \textrm{if } n &gt; 1\end{cases}" data-equation="eq:factorial_recurrence_relation">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_factorial_recurrence_relation.svg" alt="Factorial function recurrence relation">
     <br>
 </div>
 
 <!-- </equation> -->
 
 Following the convention for an [empty product][empty-product], in all definitions,
 
 <!-- <equation class="equation" label="eq:zero_factorial" align="center" raw="0! = 1" alt="Zero factorial"> -->
 
 <div class="equation" align="center" data-raw-text="0! = 1" data-equation="eq:zero_factorial">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_zero_factorial.svg" alt="Zero factorial">
     <br>
 </div>
 
 <!-- </equation> -->
 
 The [Gamma][gamma-function] function extends the [factorial function][factorial-function] for non-integer values.
 
 <!-- <equation class="equation" label="eq:factorial_function_and_gamma" align="center" raw="n! = \Gamma(n+1)" alt="Factorial function extension via the Gamma function"> -->
 
 <div class="equation" align="center" data-raw-text="n! = \Gamma(n+1)" data-equation="eq:factorial_function_and_gamma">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/factorialln/docs/img/equation_factorial_function_and_gamma.svg" alt="Factorial function extension via the Gamma function">
     <br>
 </div>
 
 <!-- </equation> -->
 
 The [factorial][factorial-function] of a **negative** integer is not defined.
 
 Evaluating the natural logarithm of [factorial function][factorial-function] is useful as the [factorial function][factorial-function] can overflow for large `n`. Thus, `factorialln( n )` is generally preferred to `ln( n! )`.
 
 </section>
 
 <!-- /.intro -->
 
 <section class="usage">
 
 ## Usage
 
 ```javascript
 var factorialln = require( '@stdlib/math/base/special/factorialln' );
 ```
 
 #### factorialln( x )
 
 Evaluates the natural logarithm of the [factorial function][factorial-function]. For input values other than negative integers, the function returns `ln( x! ) = ln( Γ(x+1) )`, where `Γ` is the [Gamma][gamma-function] function. For negative integers, the function returns `NaN`.
 
 ```javascript
 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`.
 
 ```javascript
 var v = factorialln( NaN );
 // returns NaN
 ```
 
 </section>
 
 <!-- /.usage -->
 
 <section class="examples">
 
 ## Examples
 
 <!-- eslint no-undef: "error" -->
 
 ```javascript
 var incrspace = require( '@stdlib/array/incrspace' );
 var factorialln = require( '@stdlib/math/base/special/factorialln' );
 
 var x;
 var v;
 var i;
 
 x = incrspace( -10.0, 50.0, 0.5 );
 for ( i = 0; i < x.length; i++ ) {
     v = factorialln( x[ i ] );
     console.log( 'x: %d, f(x): %d', x[ i ], v );
 }
 ```
 
 </section>
 
 <!-- /.examples -->
 
 <section class="links">
 
 [gamma-function]: https://en.wikipedia.org/wiki/Gamma_Function
 
 [factorial-function]: https://en.wikipedia.org/wiki/Factorial
 
 [empty-product]: https://en.wikipedia.org/wiki/Empty_product
 
 </section>
 
 <!-- /.links -->