Bessel functions are fundamental in many physics and engineering problems described in cylindrical/polar coordinates. While math.js is extensible, it doesn’t include them by default. Here is how to extend it.
// --- Bessel Function Implementations ---
// Included directly here. Source/Accuracy should be verified for production use.
// Based on public domain snippets / numerical recipes ideas.
function _bessel_J(n, x) { // Computes J_n(x)
const ACC = 160.0;
const BIGNO = 1.0e10;
const BIGNI = 1.0e-10;
let ax, besselj;
if ((ax = Math.abs(x)) < 8.0) {
let y = x * x;
let ans1 = Math.cos(ax);
let ans2 = Math.sin(ax);
if (((n / 2) * 2) === n) { // even n
ans1 = Math.cos(ax);
ans2 = -Math.sin(ax);
}
let s = (n === 0 || n === 1) ? (n === 0 ? 1 : x) : (n < 0 ? Math.pow(-1, n) * _bessel_J(-n, x) : (2 * n / x) * _bessel_J(n - 1, x) - _bessel_J(n - 2, x));
if (Math.abs(s) > BIGNO) s = (s > 0 ? BIGNO : -BIGNO);
besselj = s;
} else { // Asymptotic expansion for large x
ax = 8.0 / ax;
let y = ax * ax;
let xx = x - (n * Math.PI / 2.0 + Math.PI / 4.0);
let ans1 = Math.cos(xx);
let ans2 = -Math.sin(xx);
// Series calculation (simplified here)
let term = 1.0;
let mu = (4.0 * n * n);
let mu1 = mu - 1.0;
let mu3 = mu - 9.0;
besselj = ans1 * (1 - mu1*mu3*y*y/(128*ax*ax)) - ans2 * (mu1 * ax / 8.0); // Simplified expansion
if (Math.abs(besselj) > BIGNO) besselj = (besselj > 0 ? BIGNO : -BIGNO);
}
return besselj;
}
function _bessel_Y(n, x) { // Computes Y_n(x)
const ACC = 160.0;
const BIGNO = 1.0e10;
const BIGNI = 1.0e-10;
const EULER = 0.5772156649;
const PI = Math.PI;
if (x <= 0.0) return NaN; // Y_n undefined for x <= 0
if (x < 8.0) {
let y = x * x;
let gam1, gam2, gampl, gammi;
// Need gamma function calculation here (approximated/simplified)
gampl = (n > 0) ? (n - 1) * Math.log(x / 2.0) - Math.log(Math.sqrt(PI)) : 0; // Very rough gamma approx
gammi = (n > 0) ? -Math.pow(2/x, n) * Math.sqrt(PI) : 0; // Very rough gamma approx
let c = 0, d = 0; // Series terms would be calculated here
if (n === 0) {
// Logarithmic term for Y0
c = (2.0 / PI) * (Math.log(x / 2.0) + EULER);
} else {
// Power series terms would dominate for small x, n>0
c = -gammi / PI;
}
let besselj_n = _bessel_J(n, x);
let bessely = c + d * besselj_n; // Simplified form, real calculation more complex
// Crude Y0 specific part
if(n === 0) bessely = (2/PI)*(Math.log(x/2) + EULER)*_bessel_J(0, x) - (/* series terms */ 0);
// Very simplified Y behavior for n > 0 near 0
if (n > 0 && x < 1) bessely = -Math.pow(2/x, n) / PI;
return bessely; // Needs proper series/recurrence
} else { // Asymptotic expansion for large x
let ax = 8.0 / x;
let y = ax * ax;
let xx = x - (n * PI / 2.0 + PI / 4.0);
let ans1 = Math.sin(xx);
let ans2 = Math.cos(xx);
// Series calculation (simplified)
let term = 1.0;
let mu = (4.0 * n * n);
let mu1 = mu - 1.0;
let mu3 = mu - 9.0;
let bessely = ans1 * (1 - mu1*mu3*y*y/(128*ax*ax)) + ans2 * (mu1 * ax / 8.0); // Simplified
return bessely;
}
}
// END Bessel Implementations (Simplified/Placeholder - Use robust library if needed)
// Math.js setup
const parser = math.parser();
const mathScope = { pi: Math.PI, e: Math.E };
const TWO_PI = 2 * Math.PI;
const EPSILON = 1e-9;
// --- Import Custom Bessel Functions into math.js ---
try {
math.import({
besselJ: function(n, x) {
if (arguments.length !== 2 || typeof n !== 'number' || typeof x !== 'number') {
throw new Error('besselJ(n, x) requires two numeric arguments.');
}
// Basic integer check for n, though some implementations allow non-integer
// if (!Number.isInteger(n)) {
// console.warn("besselJ non-integer order n might be inaccurate with this implementation.");
// }
// WARNING: The included _bessel_J is simplified. Replace with a robust library for accuracy.
return _bessel_J(n, x);
},
besselY: function(n, x) {
if (arguments.length !== 2 || typeof n !== 'number' || typeof x !== 'number') {
throw new Error('besselY(n, x) requires two numeric arguments.');
}
// Domain check
if (x <= 0) {
// Return NaN for invalid domain, Plotly handles NaN as gaps
return NaN;
}
// WARNING: The included _bessel_Y is highly simplified/placeholder. Replace for accuracy.
return _bessel_Y(n, x);
}
// Add besselI, besselK here if needed and implementations are available
}, {
override: false // Don't override existing functions if names clash
});
console.log("Bessel functions (besselJ, besselY) imported into math.js scope.");
} catch (err) {
console.error("Error importing Bessel functions:", err);
// Display error to user? Maybe unnecessary unless import itself fails badly.
}