137 lines
3.3 KiB
Markdown
137 lines
3.3 KiB
Markdown
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@license Apache-2.0
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Copyright (c) 2019 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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# ellipe
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> Compute the [complete elliptic integral of the second kind][elliptic-integral].
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<section class="intro">
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The [complete elliptic integral of the second kind][elliptic-integral] is defined as
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<!-- <equation class="equation" label="eq:complete_elliptic_integral_second_kind" align="center" raw="E(m)=\int_0^{\pi/2} \sqrt{1 - m (\sin\theta)^2} d\theta" alt="Complete elliptic integral of the second kind."> -->
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<div class="equation" align="center" data-raw-text="E(m)=\int_0^{\pi/2} \sqrt{1 - m (\sin\theta)^2} d\theta" data-equation="eq:complete_elliptic_integral_second_kind">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@129e5a32ce2af2ed694daf2e9d4214255e60c42a/lib/node_modules/@stdlib/math/base/special/ellipe/docs/img/equation_complete_elliptic_integral_second_kind.svg" alt="Complete elliptic integral of the second kind.">
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<br>
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</div>
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<!-- </equation> -->
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where the parameter `m` is related to the modulus `k` by `m = k^2`.
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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 ellipe = require( '@stdlib/math/base/special/ellipe' );
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```
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#### ellipe( m )
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Computes the [complete elliptic integral of the second kind][elliptic-integral].
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```javascript
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var v = ellipe( 0.5 );
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// returns ~1.351
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v = ellipe( -1.0 );
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// returns ~1.910
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v = ellipe( 2.0 );
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// returns NaN
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v = ellipe( Infinity );
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// returns NaN
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v = ellipe( -Infinity );
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// returns NaN
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v = ellipe( 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="notes">
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## Notes
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- This function is valid for `-∞ < m <= 1`.
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</section>
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<!-- /.notes -->
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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 randu = require( '@stdlib/random/base/randu' );
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var ellipe = require( '@stdlib/math/base/special/ellipe' );
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var m;
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var i;
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for ( i = 0; i < 100; i++ ) {
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m = -1.0 + ( randu() * 2.0 );
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console.log( 'ellipe(%d) = %d', m, ellipe( m ) );
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}
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```
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</section>
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<!-- /.examples -->
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* * *
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<section class="references">
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## References
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- Fukushima, Toshio. 2009. "Fast computation of complete elliptic integrals and Jacobian elliptic functions." _Celestial Mechanics and Dynamical Astronomy_ 105 (4): 305. doi:[10.1007/s10569-009-9228-z][@fukushima:2009a].
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- Fukushima, Toshio. 2015. "Precise and fast computation of complete elliptic integrals by piecewise minimax rational function approximation." _Journal of Computational and Applied Mathematics_ 282 (July): 71–76. doi:[10.1016/j.cam.2014.12.038][@fukushima:2015a].
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</section>
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<!-- /.references -->
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<section class="links">
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[elliptic-integral]: https://en.wikipedia.org/wiki/Elliptic_integral
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[@fukushima:2009a]: https://doi.org/10.1007/s10569-009-9228-z
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[@fukushima:2015a]: https://doi.org/10.1016/j.cam.2014.12.038
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</section>
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<!-- /.links -->
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