'use strict';
/**
*
* This class allows the manipulation of complex numbers.
* You can pass a complex number in different formats. Either as object, double, string or two integer parameters.
*
* Object form
* { re: <real>, im: <imaginary> }
* { arg: <angle>, abs: <radius> }
* { phi: <angle>, r: <radius> }
*
* Array / Vector form
* [ real, imaginary ]
*
* Double form
* 99.3 - Single double value
*
* String form
* '23.1337' - Simple real number
* '15+3i' - a simple complex number
* '3-i' - a simple complex number
*
* Example:
*
* const c = new Complex('99.3+8i');
* c.mul({r: 3, i: 9}).div(4.9).sub(3, 2);
*
*/
const cosh = Math.cosh || function (x) {
return Math.abs(x) < 1e-9 ? 1 - x : (Math.exp(x) + Math.exp(-x)) * 0.5;
};
const sinh = Math.sinh || function (x) {
return Math.abs(x) < 1e-9 ? x : (Math.exp(x) - Math.exp(-x)) * 0.5;
};
/**
* Calculates cos(x) - 1 using Taylor series if x is small (-¼π ≤ x ≤ ¼π).
*
* @param {number} x
* @returns {number} cos(x) - 1
*/
const cosm1 = function (x) {
const b = Math.PI / 4;
if (-b > x || x > b) {
return Math.cos(x) - 1.0;
}
/* Calculate horner form of polynomial of taylor series in Q
let fac = 1, alt = 1, pol = {};
for (let i = 0; i <= 16; i++) {
fac*= i || 1;
if (i % 2 == 0) {
pol[i] = new Fraction(1, alt * fac);
alt = -alt;
}
}
console.log(new Polynomial(pol).toHorner()); // (((((((1/20922789888000x^2-1/87178291200)x^2+1/479001600)x^2-1/3628800)x^2+1/40320)x^2-1/720)x^2+1/24)x^2-1/2)x^2+1
*/
const xx = x * x;
return xx * (
xx * (
xx * (
xx * (
xx * (
xx * (
xx * (
xx / 20922789888000
- 1 / 87178291200)
+ 1 / 479001600)
- 1 / 3628800)
+ 1 / 40320)
- 1 / 720)
+ 1 / 24)
- 1 / 2);
};
const hypot = function (x, y) {
x = Math.abs(x);
y = Math.abs(y);
// Ensure `x` is the larger value
if (x < y) [x, y] = [y, x];
// If both are below the threshold, use straightforward Pythagoras
if (x < 1e8) return Math.sqrt(x * x + y * y);
// For larger values, scale to avoid overflow
y /= x;
return x * Math.sqrt(1 + y * y);
};
const parser_exit = function () {
throw SyntaxError('Invalid Param');
};
/**
* Calculates log(sqrt(a^2+b^2)) in a way to avoid overflows
*
* @param {number} a
* @param {number} b
* @returns {number}
*/
function logHypot(a, b) {
const _a = Math.abs(a);
const _b = Math.abs(b);
if (a === 0) {
return Math.log(_b);
}
if (b === 0) {
return Math.log(_a);
}
if (_a < 3000 && _b < 3000) {
return Math.log(a * a + b * b) * 0.5;
}
/* I got 4 ideas to compute this property without overflow:
*
* Testing 1000000 times with random samples for a,b ∈ [1, 1000000000] against a big decimal library to get an error estimate
*
* 1. Only eliminate the square root: (OVERALL ERROR: 3.9122483030951116e-11)
Math.log(a * a + b * b) / 2
*
*
* 2. Try to use the non-overflowing pythagoras: (OVERALL ERROR: 8.889760039210159e-10)
const fn = function(a, b) {
a = Math.abs(a);
b = Math.abs(b);
let t = Math.min(a, b);
a = Math.max(a, b);
t = t / a;
return Math.log(a) + Math.log(1 + t * t) / 2;
};
* 3. Abuse the identity cos(atan(y/x) = x / sqrt(x^2+y^2): (OVERALL ERROR: 3.4780178737037204e-10)
Math.log(a / Math.cos(Math.atan2(b, a)))
* 4. Use 3. and apply log rules: (OVERALL ERROR: 1.2014087502620896e-9)
Math.log(a) - Math.log(Math.cos(Math.atan2(b, a)))
*/
a = a * 0.5;
b = b * 0.5;
return 0.5 * Math.log(a * a + b * b) + Math.LN2;
}
const P = { 're': 0, 'im': 0 };
const parse = function (a, b) {
const z = P;
if (a === undefined || a === null) {
z['re'] =
z['im'] = 0;
} else if (b !== undefined) {
z['re'] = a;
z['im'] = b;
} else
switch (typeof a) {
case 'object':
if ('im' in a && 're' in a) {
z['re'] = a['re'];
z['im'] = a['im'];
} else if ('abs' in a && 'arg' in a) {
if (!isFinite(a['abs']) && isFinite(a['arg'])) {
return Complex['INFINITY'];
}
z['re'] = a['abs'] * Math.cos(a['arg']);
z['im'] = a['abs'] * Math.sin(a['arg']);
} else if ('r' in a && 'phi' in a) {
if (!isFinite(a['r']) && isFinite(a['phi'])) {
return Complex['INFINITY'];
}
z['re'] = a['r'] * Math.cos(a['phi']);
z['im'] = a['r'] * Math.sin(a['phi']);
} else if (a.length === 2) { // Quick array check
z['re'] = a[0];
z['im'] = a[1];
} else {
parser_exit();
}
break;
case 'string':
z['im'] = /* void */
z['re'] = 0;
const tokens = a.replace(/_/g, '')
.match(/\d+\.?\d*e[+-]?\d+|\d+\.?\d*|\.\d+|./g);
let plus = 1;
let minus = 0;
if (tokens === null) {
parser_exit();
}
for (let i = 0; i < tokens.length; i++) {
const c = tokens[i];
if (c === ' ' || c === '\t' || c === '\n') {
/* void */
} else if (c === '+') {
plus++;
} else if (c === '-') {
minus++;
} else if (c === 'i' || c === 'I') {
if (plus + minus === 0) {
parser_exit();
}
if (tokens[i + 1] !== ' ' && !isNaN(tokens[i + 1])) {
z['im'] += parseFloat((minus % 2 ? '-' : '') + tokens[i + 1]);
i++;
} else {
z['im'] += parseFloat((minus % 2 ? '-' : '') + '1');
}
plus = minus = 0;
} else {
if (plus + minus === 0 || isNaN(c)) {
parser_exit();
}
if (tokens[i + 1] === 'i' || tokens[i + 1] === 'I') {
z['im'] += parseFloat((minus % 2 ? '-' : '') + c);
i++;
} else {
z['re'] += parseFloat((minus % 2 ? '-' : '') + c);
}
plus = minus = 0;
}
}
// Still something on the stack
if (plus + minus > 0) {
parser_exit();
}
break;
case 'number':
z['im'] = 0;
z['re'] = a;
break;
default:
parser_exit();
}
if (isNaN(z['re']) || isNaN(z['im'])) {
// If a calculation is NaN, we treat it as NaN and don't throw
//parser_exit();
}
return z;
};
/**
* @constructor
* @returns {Complex}
*/
function Complex(a, b) {
if (!(this instanceof Complex)) {
return new Complex(a, b);
}
const z = parse(a, b);
this['re'] = z['re'];
this['im'] = z['im'];
}
Complex.prototype = {
're': 0,
'im': 0,
/**
* Calculates the sign of a complex number, which is a normalized complex
*
* @returns {Complex}
*/
'sign': function () {
const abs = hypot(this['re'], this['im']);
return new Complex(
this['re'] / abs,
this['im'] / abs);
},
/**
* Adds two complex numbers
*
* @returns {Complex}
*/
'add': function (a, b) {
const z = parse(a, b);
const tInfin = this['isInfinite']();
const zInfin = !(isFinite(z['re']) && isFinite(z['im']));
if (tInfin || zInfin) {
if (tInfin && zInfin) {
// Infinity + Infinity = NaN
return Complex['NAN'];
}
// Infinity + z = Infinity { where z != Infinity }
return Complex['INFINITY'];
}
return new Complex(
this['re'] + z['re'],
this['im'] + z['im']);
},
/**
* Subtracts two complex numbers
*
* @returns {Complex}
*/
'sub': function (a, b) {
const z = parse(a, b);
const tInfin = this['isInfinite']();
const zInfin = !(isFinite(z['re']) && isFinite(z['im']));
if (tInfin || zInfin) {
if (tInfin && zInfin) {
// Infinity - Infinity = NaN
return Complex['NAN'];
}
// Infinity - z = Infinity { where z != Infinity }
return Complex['INFINITY'];
}
return new Complex(
this['re'] - z['re'],
this['im'] - z['im']);
},
/**
* Multiplies two complex numbers
*
* @returns {Complex}
*/
'mul': function (a, b) {
const z = parse(a, b);
const tInfin = this['isInfinite']();
const zInfin = !(isFinite(z['re']) && isFinite(z['im']));
const tIsZero = this['re'] === 0 && this['im'] === 0;
const zIsZero = z['re'] === 0 && z['im'] === 0;
// Infinity * 0 = NaN
if (tInfin && zIsZero || zInfin && tIsZero) {
return Complex['NAN'];
}
// Infinity * z = Infinity { where z != 0 }
if (tInfin || zInfin) {
return Complex['INFINITY'];
}
// Shortcut for real values
if (z['im'] === 0 && this['im'] === 0) {
return new Complex(this['re'] * z['re'], 0);
}
return new Complex(
this['re'] * z['re'] - this['im'] * z['im'],
this['re'] * z['im'] + this['im'] * z['re']);
},
/**
* Divides two complex numbers
*
* @returns {Complex}
*/
'div': function (a, b) {
const z = parse(a, b);
const tInfin = this['isInfinite']();
const zInfin = !(isFinite(z['re']) && isFinite(z['im']));
const tIsZero = this['re'] === 0 && this['im'] === 0;
const zIsZero = z['re'] === 0 && z['im'] === 0;
// 0 / 0 = NaN and Infinity / Infinity = NaN
if (tIsZero && zIsZero || tInfin && zInfin) {
return Complex['NAN'];
}
// Infinity / 0 = Infinity
if (zIsZero || tInfin) {
return Complex['INFINITY'];
}
// 0 / Infinity = 0
if (tIsZero || zInfin) {
return Complex['ZERO'];
}
if (0 === z['im']) {
// Divisor is real
return new Complex(this['re'] / z['re'], this['im'] / z['re']);
}
if (Math.abs(z['re']) < Math.abs(z['im'])) {
const x = z['re'] / z['im'];
const t = z['re'] * x + z['im'];
return new Complex(
(this['re'] * x + this['im']) / t,
(this['im'] * x - this['re']) / t);
} else {
const x = z['im'] / z['re'];
const t = z['im'] * x + z['re'];
return new Complex(
(this['re'] + this['im'] * x) / t,
(this['im'] - this['re'] * x) / t);
}
},
/**
* Calculate the power of two complex numbers
*
* @returns {Complex}
*/
'pow': function (a, b) {
const z = parse(a, b);
const tIsZero = this['re'] === 0 && this['im'] === 0;
const zIsZero = z['re'] === 0 && z['im'] === 0;
if (zIsZero) {
return Complex['ONE'];
}
// If the exponent is real
if (z['im'] === 0) {
if (this['im'] === 0 && this['re'] > 0) {
return new Complex(Math.pow(this['re'], z['re']), 0);
} else if (this['re'] === 0) { // If base is fully imaginary
switch ((z['re'] % 4 + 4) % 4) {
case 0:
return new Complex(Math.pow(this['im'], z['re']), 0);
case 1:
return new Complex(0, Math.pow(this['im'], z['re']));
case 2:
return new Complex(-Math.pow(this['im'], z['re']), 0);
case 3:
return new Complex(0, -Math.pow(this['im'], z['re']));
}
}
}
/* I couldn't find a good formula, so here is a derivation and optimization
*
* z_1^z_2 = (a + bi)^(c + di)
* = exp((c + di) * log(a + bi)
* = pow(a^2 + b^2, (c + di) / 2) * exp(i(c + di)atan2(b, a))
* =>...
* Re = (pow(a^2 + b^2, c / 2) * exp(-d * atan2(b, a))) * cos(d * log(a^2 + b^2) / 2 + c * atan2(b, a))
* Im = (pow(a^2 + b^2, c / 2) * exp(-d * atan2(b, a))) * sin(d * log(a^2 + b^2) / 2 + c * atan2(b, a))
*
* =>...
* Re = exp(c * log(sqrt(a^2 + b^2)) - d * atan2(b, a)) * cos(d * log(sqrt(a^2 + b^2)) + c * atan2(b, a))
* Im = exp(c * log(sqrt(a^2 + b^2)) - d * atan2(b, a)) * sin(d * log(sqrt(a^2 + b^2)) + c * atan2(b, a))
*
* =>
* Re = exp(c * logsq2 - d * arg(z_1)) * cos(d * logsq2 + c * arg(z_1))
* Im = exp(c * logsq2 - d * arg(z_1)) * sin(d * logsq2 + c * arg(z_1))
*
*/
if (tIsZero && z['re'] > 0) { // Same behavior as Wolframalpha, Zero if real part is zero
return Complex['ZERO'];
}
const arg = Math.atan2(this['im'], this['re']);
const loh = logHypot(this['re'], this['im']);
let re = Math.exp(z['re'] * loh - z['im'] * arg);
let im = z['im'] * loh + z['re'] * arg;
return new Complex(
re * Math.cos(im),
re * Math.sin(im));
},
/**
* Calculate the complex square root
*
* @returns {Complex}
*/
'sqrt': function () {
const a = this['re'];
const b = this['im'];
if (b === 0) {
// Real number case
if (a >= 0) {
return new Complex(Math.sqrt(a), 0);
} else {
return new Complex(0, Math.sqrt(-a));
}
}
const r = hypot(a, b);
let re = Math.sqrt(0.5 * (r + Math.abs(a))); // sqrt(2x) / 2 = sqrt(x / 2)
let im = Math.abs(b) / (2 * re);
if (a >= 0) {
return new Complex(re, b < 0 ? -im : im);
} else {
return new Complex(im, b < 0 ? -re : re);
}
},
/**
* Calculate the complex exponent
*
* @returns {Complex}
*/
'exp': function () {
const er = Math.exp(this['re']);
if (this['im'] === 0) {
return new Complex(er, 0);
}
return new Complex(
er * Math.cos(this['im']),
er * Math.sin(this['im']));
},
/**
* Calculate the complex exponent and subtracts one.
*
* This may be more accurate than `Complex(x).exp().sub(1)` if
* `x` is small.
*
* @returns {Complex}
*/
'expm1': function () {
/**
* exp(a + i*b) - 1
= exp(a) * (cos(b) + j*sin(b)) - 1
= expm1(a)*cos(b) + cosm1(b) + j*exp(a)*sin(b)
*/
const a = this['re'];
const b = this['im'];
return new Complex(
Math.expm1(a) * Math.cos(b) + cosm1(b),
Math.exp(a) * Math.sin(b));
},
/**
* Calculate the natural log
*
* @returns {Complex}
*/
'log': function () {
const a = this['re'];
const b = this['im'];
if (b === 0 && a > 0) {
return new Complex(Math.log(a), 0);
}
return new Complex(
logHypot(a, b),
Math.atan2(b, a));
},
/**
* Calculate the magnitude of the complex number
*
* @returns {number}
*/
'abs': function () {
return hypot(this['re'], this['im']);
},
/**
* Calculate the angle of the complex number
*
* @returns {number}
*/
'arg': function () {
return Math.atan2(this['im'], this['re']);
},
/**
* Calculate the sine of the complex number
*
* @returns {Complex}
*/
'sin': function () {
// sin(z) = ( e^iz - e^-iz ) / 2i
// = sin(a)cosh(b) + i cos(a)sinh(b)
const a = this['re'];
const b = this['im'];
return new Complex(
Math.sin(a) * cosh(b),
Math.cos(a) * sinh(b));
},
/**
* Calculate the cosine
*
* @returns {Complex}
*/
'cos': function () {
// cos(z) = ( e^iz + e^-iz ) / 2
// = cos(a)cosh(b) - i sin(a)sinh(b)
const a = this['re'];
const b = this['im'];
return new Complex(
Math.cos(a) * cosh(b),
-Math.sin(a) * sinh(b));
},
/**
* Calculate the tangent
*
* @returns {Complex}
*/
'tan': function () {
// tan(z) = sin(z) / cos(z)
// = ( e^iz - e^-iz ) / ( i( e^iz + e^-iz ) )
// = ( e^2iz - 1 ) / i( e^2iz + 1 )
// = ( sin(2a) + i sinh(2b) ) / ( cos(2a) + cosh(2b) )
const a = 2 * this['re'];
const b = 2 * this['im'];
const d = Math.cos(a) + cosh(b);
return new Complex(
Math.sin(a) / d,
sinh(b) / d);
},
/**
* Calculate the cotangent
*
* @returns {Complex}
*/
'cot': function () {
// cot(c) = i(e^(ci) + e^(-ci)) / (e^(ci) - e^(-ci))
const a = 2 * this['re'];
const b = 2 * this['im'];
const d = Math.cos(a) - cosh(b);
return new Complex(
-Math.sin(a) / d,
sinh(b) / d);
},
/**
* Calculate the secant
*
* @returns {Complex}
*/
'sec': function () {
// sec(c) = 2 / (e^(ci) + e^(-ci))
const a = this['re'];
const b = this['im'];
const d = 0.5 * cosh(2 * b) + 0.5 * Math.cos(2 * a);
return new Complex(
Math.cos(a) * cosh(b) / d,
Math.sin(a) * sinh(b) / d);
},
/**
* Calculate the cosecans
*
* @returns {Complex}
*/
'csc': function () {
// csc(c) = 2i / (e^(ci) - e^(-ci))
const a = this['re'];
const b = this['im'];
const d = 0.5 * cosh(2 * b) - 0.5 * Math.cos(2 * a);
return new Complex(
Math.sin(a) * cosh(b) / d,
-Math.cos(a) * sinh(b) / d);
},
/**
* Calculate the complex arcus sinus
*
* @returns {Complex}
*/
'asin': function () {
// asin(c) = -i * log(ci + sqrt(1 - c^2))
const a = this['re'];
const b = this['im'];
const t1 = new Complex(
b * b - a * a + 1,
-2 * a * b)['sqrt']();
const t2 = new Complex(
t1['re'] - b,
t1['im'] + a)['log']();
return new Complex(t2['im'], -t2['re']);
},
/**
* Calculate the complex arcus cosinus
*
* @returns {Complex}
*/
'acos': function () {
// acos(c) = i * log(c - i * sqrt(1 - c^2))
const a = this['re'];
const b = this['im'];
const t1 = new Complex(
b * b - a * a + 1,
-2 * a * b)['sqrt']();
const t2 = new Complex(
t1['re'] - b,
t1['im'] + a)['log']();
return new Complex(Math.PI / 2 - t2['im'], t2['re']);
},
/**
* Calculate the complex arcus tangent
*
* @returns {Complex}
*/
'atan': function () {
// atan(c) = i / 2 log((i + x) / (i - x))
const a = this['re'];
const b = this['im'];
if (a === 0) {
if (b === 1) {
return new Complex(0, Infinity);
}
if (b === -1) {
return new Complex(0, -Infinity);
}
}
const d = a * a + (1.0 - b) * (1.0 - b);
const t1 = new Complex(
(1 - b * b - a * a) / d,
-2 * a / d).log();
return new Complex(-0.5 * t1['im'], 0.5 * t1['re']);
},
/**
* Calculate the complex arcus cotangent
*
* @returns {Complex}
*/
'acot': function () {
// acot(c) = i / 2 log((c - i) / (c + i))
const a = this['re'];
const b = this['im'];
if (b === 0) {
return new Complex(Math.atan2(1, a), 0);
}
const d = a * a + b * b;
return (d !== 0)
? new Complex(
a / d,
-b / d).atan()
: new Complex(
(a !== 0) ? a / 0 : 0,
(b !== 0) ? -b / 0 : 0).atan();
},
/**
* Calculate the complex arcus secant
*
* @returns {Complex}
*/
'asec': function () {
// asec(c) = -i * log(1 / c + sqrt(1 - i / c^2))
const a = this['re'];
const b = this['im'];
if (a === 0 && b === 0) {
return new Complex(0, Infinity);
}
const d = a * a + b * b;
return (d !== 0)
? new Complex(
a / d,
-b / d).acos()
: new Complex(
(a !== 0) ? a / 0 : 0,
(b !== 0) ? -b / 0 : 0).acos();
},
/**
* Calculate the complex arcus cosecans
*
* @returns {Complex}
*/
'acsc': function () {
// acsc(c) = -i * log(i / c + sqrt(1 - 1 / c^2))
const a = this['re'];
const b = this['im'];
if (a === 0 && b === 0) {
return new Complex(Math.PI / 2, Infinity);
}
const d = a * a + b * b;
return (d !== 0)
? new Complex(
a / d,
-b / d).asin()
: new Complex(
(a !== 0) ? a / 0 : 0,
(b !== 0) ? -b / 0 : 0).asin();
},
/**
* Calculate the complex sinh
*
* @returns {Complex}
*/
'sinh': function () {
// sinh(c) = (e^c - e^-c) / 2
const a = this['re'];
const b = this['im'];
return new Complex(
sinh(a) * Math.cos(b),
cosh(a) * Math.sin(b));
},
/**
* Calculate the complex cosh
*
* @returns {Complex}
*/
'cosh': function () {
// cosh(c) = (e^c + e^-c) / 2
const a = this['re'];
const b = this['im'];
return new Complex(
cosh(a) * Math.cos(b),
sinh(a) * Math.sin(b));
},
/**
* Calculate the complex tanh
*
* @returns {Complex}
*/
'tanh': function () {
// tanh(c) = (e^c - e^-c) / (e^c + e^-c)
const a = 2 * this['re'];
const b = 2 * this['im'];
const d = cosh(a) + Math.cos(b);
return new Complex(
sinh(a) / d,
Math.sin(b) / d);
},
/**
* Calculate the complex coth
*
* @returns {Complex}
*/
'coth': function () {
// coth(c) = (e^c + e^-c) / (e^c - e^-c)
const a = 2 * this['re'];
const b = 2 * this['im'];
const d = cosh(a) - Math.cos(b);
return new Complex(
sinh(a) / d,
-Math.sin(b) / d);
},
/**
* Calculate the complex coth
*
* @returns {Complex}
*/
'csch': function () {
// csch(c) = 2 / (e^c - e^-c)
const a = this['re'];
const b = this['im'];
const d = Math.cos(2 * b) - cosh(2 * a);
return new Complex(
-2 * sinh(a) * Math.cos(b) / d,
2 * cosh(a) * Math.sin(b) / d);
},
/**
* Calculate the complex sech
*
* @returns {Complex}
*/
'sech': function () {
// sech(c) = 2 / (e^c + e^-c)
const a = this['re'];
const b = this['im'];
const d = Math.cos(2 * b) + cosh(2 * a);
return new Complex(
2 * cosh(a) * Math.cos(b) / d,
-2 * sinh(a) * Math.sin(b) / d);
},
/**
* Calculate the complex asinh
*
* @returns {Complex}
*/
'asinh': function () {
// asinh(c) = log(c + sqrt(c^2 + 1))
let tmp = this['im'];
this['im'] = -this['re'];
this['re'] = tmp;
const res = this['asin']();
this['re'] = -this['im'];
this['im'] = tmp;
tmp = res['re'];
res['re'] = -res['im'];
res['im'] = tmp;
return res;
},
/**
* Calculate the complex acosh
*
* @returns {Complex}
*/
'acosh': function () {
// acosh(c) = log(c + sqrt(c^2 - 1))
const res = this['acos']();
if (res['im'] <= 0) {
const tmp = res['re'];
res['re'] = -res['im'];
res['im'] = tmp;
} else {
const tmp = res['im'];
res['im'] = -res['re'];
res['re'] = tmp;
}
return res;
},
/**
* Calculate the complex atanh
*
* @returns {Complex}
*/
'atanh': function () {
// atanh(c) = log((1+c) / (1-c)) / 2
const a = this['re'];
const b = this['im'];
const noIM = a > 1 && b === 0;
const oneMinus = 1 - a;
const onePlus = 1 + a;
const d = oneMinus * oneMinus + b * b;
const x = (d !== 0)
? new Complex(
(onePlus * oneMinus - b * b) / d,
(b * oneMinus + onePlus * b) / d)
: new Complex(
(a !== -1) ? (a / 0) : 0,
(b !== 0) ? (b / 0) : 0);
const temp = x['re'];
x['re'] = logHypot(x['re'], x['im']) / 2;
x['im'] = Math.atan2(x['im'], temp) / 2;
if (noIM) {
x['im'] = -x['im'];
}
return x;
},
/**
* Calculate the complex acoth
*
* @returns {Complex}
*/
'acoth': function () {
// acoth(c) = log((c+1) / (c-1)) / 2
const a = this['re'];
const b = this['im'];
if (a === 0 && b === 0) {
return new Complex(0, Math.PI / 2);
}
const d = a * a + b * b;
return (d !== 0)
? new Complex(
a / d,
-b / d).atanh()
: new Complex(
(a !== 0) ? a / 0 : 0,
(b !== 0) ? -b / 0 : 0).atanh();
},
/**
* Calculate the complex acsch
*
* @returns {Complex}
*/
'acsch': function () {
// acsch(c) = log((1+sqrt(1+c^2))/c)
const a = this['re'];
const b = this['im'];
if (b === 0) {
return new Complex(
(a !== 0)
? Math.log(a + Math.sqrt(a * a + 1))
: Infinity, 0);
}
const d = a * a + b * b;
return (d !== 0)
? new Complex(
a / d,
-b / d).asinh()
: new Complex(
(a !== 0) ? a / 0 : 0,
(b !== 0) ? -b / 0 : 0).asinh();
},
/**
* Calculate the complex asech
*
* @returns {Complex}
*/
'asech': function () {
// asech(c) = log((1+sqrt(1-c^2))/c)
const a = this['re'];
const b = this['im'];
if (this['isZero']()) {
return Complex['INFINITY'];
}
const d = a * a + b * b;
return (d !== 0)
? new Complex(
a / d,
-b / d).acosh()
: new Complex(
(a !== 0) ? a / 0 : 0,
(b !== 0) ? -b / 0 : 0).acosh();
},
/**
* Calculate the complex inverse 1/z
*
* @returns {Complex}
*/
'inverse': function () {
// 1 / 0 = Infinity and 1 / Infinity = 0
if (this['isZero']()) {
return Complex['INFINITY'];
}
if (this['isInfinite']()) {
return Complex['ZERO'];
}
const a = this['re'];
const b = this['im'];
const d = a * a + b * b;
return new Complex(a / d, -b / d);
},
/**
* Returns the complex conjugate
*
* @returns {Complex}
*/
'conjugate': function () {
return new Complex(this['re'], -this['im']);
},
/**
* Gets the negated complex number
*
* @returns {Complex}
*/
'neg': function () {
return new Complex(-this['re'], -this['im']);
},
/**
* Ceils the actual complex number
*
* @returns {Complex}
*/
'ceil': function (places) {
places = Math.pow(10, places || 0);
return new Complex(
Math.ceil(this['re'] * places) / places,
Math.ceil(this['im'] * places) / places);
},
/**
* Floors the actual complex number
*
* @returns {Complex}
*/
'floor': function (places) {
places = Math.pow(10, places || 0);
return new Complex(
Math.floor(this['re'] * places) / places,
Math.floor(this['im'] * places) / places);
},
/**
* Ceils the actual complex number
*
* @returns {Complex}
*/
'round': function (places) {
places = Math.pow(10, places || 0);
return new Complex(
Math.round(this['re'] * places) / places,
Math.round(this['im'] * places) / places);
},
/**
* Compares two complex numbers
*
* **Note:** new Complex(Infinity).equals(Infinity) === false
*
* @returns {boolean}
*/
'equals': function (a, b) {
const z = parse(a, b);
return Math.abs(z['re'] - this['re']) <= Complex['EPSILON'] &&
Math.abs(z['im'] - this['im']) <= Complex['EPSILON'];
},
/**
* Clones the actual object
*
* @returns {Complex}
*/
'clone': function () {
return new Complex(this['re'], this['im']);
},
/**
* Gets a string of the actual complex number
*
* @returns {string}
*/
'toString': function () {
let a = this['re'];
let b = this['im'];
let ret = "";
if (this['isNaN']()) {
return 'NaN';
}
if (this['isInfinite']()) {
return 'Infinity';
}
if (Math.abs(a) < Complex['EPSILON']) {
a = 0;
}
if (Math.abs(b) < Complex['EPSILON']) {
b = 0;
}
// If is real number
if (b === 0) {
return ret + a;
}
if (a !== 0) {
ret += a;
ret += " ";
if (b < 0) {
b = -b;
ret += "-";
} else {
ret += "+";
}
ret += " ";
} else if (b < 0) {
b = -b;
ret += "-";
}
if (1 !== b) { // b is the absolute imaginary part
ret += b;
}
return ret + "i";
},
/**
* Returns the actual number as a vector
*
* @returns {Array}
*/
'toVector': function () {
return [this['re'], this['im']];
},
/**
* Returns the actual real value of the current object
*
* @returns {number|null}
*/
'valueOf': function () {
if (this['im'] === 0) {
return this['re'];
}
return null;
},
/**
* Determines whether a complex number is not on the Riemann sphere.
*
* @returns {boolean}
*/
'isNaN': function () {
return isNaN(this['re']) || isNaN(this['im']);
},
/**
* Determines whether or not a complex number is at the zero pole of the
* Riemann sphere.
*
* @returns {boolean}
*/
'isZero': function () {
return this['im'] === 0 && this['re'] === 0;
},
/**
* Determines whether a complex number is not at the infinity pole of the
* Riemann sphere.
*
* @returns {boolean}
*/
'isFinite': function () {
return isFinite(this['re']) && isFinite(this['im']);
},
/**
* Determines whether or not a complex number is at the infinity pole of the
* Riemann sphere.
*
* @returns {boolean}
*/
'isInfinite': function () {
return !this['isFinite']();
}
};
Complex['ZERO'] = new Complex(0, 0);
Complex['ONE'] = new Complex(1, 0);
Complex['I'] = new Complex(0, 1);
Complex['PI'] = new Complex(Math.PI, 0);
Complex['E'] = new Complex(Math.E, 0);
Complex['INFINITY'] = new Complex(Infinity, Infinity);
Complex['NAN'] = new Complex(NaN, NaN);
Complex['EPSILON'] = 1e-15;
export {
Complex as default, Complex
};