bessel/README.md

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# bessel
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Pure-JS implementation of Bessel functions J,Y,I,K (for the browser and nodejs).
Emphasis on correctness and performance for integer order.
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The standard notation is used here:
- `J` is the Bessel function of the first kind
- `Y` is the Bessel function of the second kind
- `I` is the modified Bessel function of the first kind
- `K` is the modified Bessel function of the second kind
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## Installation
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With [npm](https://www.npmjs.org/package/bessel):
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```bash
$ npm install bessel
```
In the browser:
```html
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<script src="bessel.js"></script>
```
The browser exposes a variable `BESSEL`
The script will manipulate `module.exports` if available (e.g. in a CommonJS
`require` context). This is not always desirable. To prevent the behavior,
define `DO_NOT_EXPORT_BESSEL`
## Usage
In all cases, the relevant function takes two arguments (`value`, `order`).
The return value is a JS number. `NaN` signals an error in calculation.
- `BESSEL.besselj(x, n)` computes `J_{n}(x)`
- `BESSEL.bessely(x, n)` computes `Y_{n}(x)`
- `BESSEL.besseli(x, n)` computes `I_{n}(x)`
- `BESSEL.besselk(x, n)` computes `K_{n}(x)`
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For example:
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```js
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// var BESSEL = require('bessel'); // uncomment this line if in node
BESSEL.besselj(1.5,0) // 0.5118276712499389
BESSEL.bessely(1.5,0) // 0.38244892476502895
BESSEL.besseli(1.5,0) // 1.6467232021476754
BESSEL.besselk(1.5,0) // 0.2138055693236539
var Y = BESSEL.bessely
Y(Math.PI, 5) + Y(Math.PI, 3) - (2 * 4 / Math.PI) * Y(Math.PI, 4) // 0
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```
## Testing
`make test` will run the nodejs-based test.
To generate the `excel.tsv` test cases, make a 6-column Excel Sheet:
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| x | n | `BESSELI` | `BESSELJ` | `BESSELK` | `BESSELY` |
|---|---|:--------------:|:--------------:|:--------------:|:--------------:|
| x | n |`BESSELI(A1,B1)`|`BESSELJ(A1,B1)`|`BESSELK(A1,B1)`|`BESSELY(A1,B1)`|
To generate the `mma.tsv` test cases, use the Mathematica Bessel Functions:
```mathematica
(* Bessel_[value, order] *)
F[x_,n_]:={x/2,n,BesselI[n,x/2], BesselJ[n,x/2], BesselK[n,x/2], BesselY[n,x/2]}
Do[Print[ExportString[N[F[x,n],10],"csv"]], {n,1,3}, {x,1,20} ]
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```
Note: Each function follows Excel semantics `(value, order)`. Other platforms
like Mathematica reverse the argument order.
## License
Please consult the attached LICENSE file for details. All rights not explicitly
granted by the Apache 2.0 License are reserved by the Original Author.
## Badges
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