114 lines
2.6 KiB
Markdown
114 lines
2.6 KiB
Markdown
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@license Apache-2.0
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Copyright (c) 2018 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# Riemann Zeta Function
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> [Riemann zeta][zeta-function] function.
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<section class="intro">
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The [Riemann zeta][zeta-function] function is the [analytic continuation][analytic-continuation] of the infinite series
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<!-- <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"> -->
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<div class="equation" align="center" data-raw-text="\zeta(s) =\sum_{k=1}^\infty\frac{1}{k^s}" data-equation="eq:riemann_zeta_function">
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<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">
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<br>
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</div>
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<!-- </equation> -->
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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`.
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var zeta = require( '@stdlib/math/base/special/riemann-zeta' );
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```
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#### zeta( s )
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Evaluates the [Riemann zeta][zeta-function] function as a function of a real variable `s` (i.e., `t = 0`).
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```javascript
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var v = zeta( 1.1 );
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// returns ~10.584
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v = zeta( -4.0 );
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// returns 0.0
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v = zeta( 70.0 );
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// returns 1.0
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v = zeta( 0.5 );
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// returns ~-1.46
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v = zeta( 1.0 ); // pole
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// returns NaN
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v = zeta( NaN );
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// returns NaN
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```
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</section>
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<!-- /.usage -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var linspace = require( '@stdlib/array/linspace' );
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var zeta = require( '@stdlib/math/base/special/riemann-zeta' );
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var s;
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var v;
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var i;
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s = linspace( -50.0, 50.0, 200 );
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for ( i = 0; i < s.length; i++ ) {
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v = zeta( s[ i ] );
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console.log( 's: %d, ζ(s): %d', s[ i ], v );
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}
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```
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</section>
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<!-- /.examples -->
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<section class="links">
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[zeta-function]: https://en.wikipedia.org/wiki/Riemann_zeta_function
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[analytic-continuation]: https://en.wikipedia.org/wiki/Analytic_continuation
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</section>
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<!-- /.links -->
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