bessel

Pure-JS implementation of Bessel functions J,Y,I,K (for the browser and nodejs). Emphasis on correctness and performance for integer order.

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

Installation

With npm:

$ npm install bessel

In the browser:

<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)

For example:

> // 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

Testing

make test will run the nodejs-based test.

To generate the excel.tsv test cases, make a 6-column Excel Sheet:

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:

(* 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} ]

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 MIT License are reserved by the Original Author.

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