time-to-botec

Benchmark sampling in different programming languages
Log | Files | Refs | README

README.md (2666B)

      1 <!--
      2 
      3 @license Apache-2.0
      4 
      5 Copyright (c) 2018 The Stdlib Authors.
      6 
      7 Licensed under the Apache License, Version 2.0 (the "License");
      8 you may not use this file except in compliance with the License.
      9 You may obtain a copy of the License at
     10 
     11    http://www.apache.org/licenses/LICENSE-2.0
     12 
     13 Unless required by applicable law or agreed to in writing, software
     14 distributed under the License is distributed on an "AS IS" BASIS,
     15 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
     16 See the License for the specific language governing permissions and
     17 limitations under the License.
     18 
     19 -->
     20 
     21 # Riemann Zeta Function
     22 
     23 > [Riemann zeta][zeta-function] function.
     24 
     25 <section class="intro">
     26 
     27 The [Riemann zeta][zeta-function] function is the [analytic continuation][analytic-continuation] of the infinite series
     28 
     29 <!-- <equation class="equation" label="eq:riemann_zeta_function" align="center" raw="\zeta(s) =\sum_{k=1}^\infty\frac{1}{k^s}" alt="Riemann zeta function"> -->
     30 
     31 <div class="equation" align="center" data-raw-text="\zeta(s) =\sum_{k=1}^\infty\frac{1}{k^s}" data-equation="eq:riemann_zeta_function">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/riemann-zeta/docs/img/equation_riemann_zeta_function.svg" alt="Riemann zeta function">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 where `s` is a complex variable equal to `σ + ti`. The series is only convergent when the real part of `s`, `σ`, is greater than `1`.
     39 
     40 </section>
     41 
     42 <!-- /.intro -->
     43 
     44 <section class="usage">
     45 
     46 ## Usage
     47 
     48 ```javascript
     49 var zeta = require( '@stdlib/math/base/special/riemann-zeta' );
     50 ```
     51 
     52 #### zeta( s )
     53 
     54 Evaluates the [Riemann zeta][zeta-function] function as a function of a real variable `s` (i.e., `t = 0`).
     55 
     56 ```javascript
     57 var v = zeta( 1.1 );
     58 // returns ~10.584
     59 
     60 v = zeta( -4.0 );
     61 // returns 0.0
     62 
     63 v = zeta( 70.0 );
     64 // returns 1.0
     65 
     66 v = zeta( 0.5 );
     67 // returns ~-1.46
     68 
     69 v = zeta( 1.0 ); // pole
     70 // returns NaN
     71 
     72 v = zeta( NaN );
     73 // returns NaN
     74 ```
     75 
     76 </section>
     77 
     78 <!-- /.usage -->
     79 
     80 <section class="examples">
     81 
     82 ## Examples
     83 
     84 <!-- eslint no-undef: "error" -->
     85 
     86 ```javascript
     87 var linspace = require( '@stdlib/array/linspace' );
     88 var zeta = require( '@stdlib/math/base/special/riemann-zeta' );
     89 
     90 var s;
     91 var v;
     92 var i;
     93 
     94 s = linspace( -50.0, 50.0, 200 );
     95 for ( i = 0; i < s.length; i++ ) {
     96     v = zeta( s[ i ] );
     97     console.log( 's: %d, ζ(s): %d', s[ i ], v );
     98 }
     99 ```
    100 
    101 </section>
    102 
    103 <!-- /.examples -->
    104 
    105 <section class="links">
    106 
    107 [zeta-function]: https://en.wikipedia.org/wiki/Riemann_zeta_function
    108 
    109 [analytic-continuation]: https://en.wikipedia.org/wiki/Analytic_continuation
    110 
    111 </section>
    112 
    113 <!-- /.links -->