complex.js

What is complex.js?

The complex.js npm package is a library for complex number arithmetic in JavaScript. It provides a comprehensive set of functions to perform operations on complex numbers, including basic arithmetic, trigonometric functions, and more.

What are complex.js's main functionalities?

Basic Arithmetic

This feature allows you to perform basic arithmetic operations such as addition, subtraction, multiplication, and division on complex numbers.

const Complex = require('complex.js');
const a = new Complex(2, 3);
const b = new Complex(1, 1);
const sum = a.add(b);
console.log(sum.toString()); // '3 + 4i'

Trigonometric Functions

This feature provides trigonometric functions like sine, cosine, and tangent for complex numbers.

const Complex = require('complex.js');
const a = new Complex(1, 1);
const sinA = a.sin();
console.log(sinA.toString()); // '1.2984575814159773 + 0.6349639147847361i'

Exponential and Logarithmic Functions

This feature allows you to compute the exponential and logarithmic functions of complex numbers.

const Complex = require('complex.js');
const a = new Complex(1, 1);
const expA = a.exp();
console.log(expA.toString()); // '1.4686939399158851 + 2.2873552871788423i'

Polar Coordinates

This feature allows you to convert complex numbers to and from polar coordinates.

const Complex = require('complex.js');
const a = new Complex(1, 1);
const polar = a.toPolar();
console.log(polar); // { r: 1.4142135623730951, phi: 0.7853981633974483 }

Other packages similar to complex.js

mathjs

Math.js is an extensive math library for JavaScript and Node.js. It supports complex numbers, matrices, units, and many other mathematical functions. Compared to complex.js, math.js offers a broader range of mathematical functionalities but may be more complex to use for operations specifically focused on complex numbers.

numeric

Numeric is a library for numerical analysis in JavaScript. It includes support for complex numbers, matrix operations, and other numerical methods. While it provides similar functionalities for complex numbers, it is more focused on numerical analysis and matrix computations.

complex-number

Complex-number is a lightweight library for complex number arithmetic in JavaScript. It offers basic operations and some advanced functions for complex numbers. It is simpler and more lightweight compared to complex.js but may lack some of the advanced features.

README

Complex.js - ℂ in JavaScript

Complex.js is a well tested JavaScript library to work with complex number arithmetic in JavaScript. It implements every elementary complex number manipulation function and the API is intentionally similar to Fraction.js. Furthermore, it's the basis of Polynomial.js and Math.js.

Examples

let Complex = require('complex.js');

let c = new Complex("99.3+8i");
c.mul({re: 3, im: 9}).div(4.9).sub(3, 2);

A classical use case for complex numbers is solving quadratic equations ax² + bx + c = 0 for all a, b, c ∈ ℝ:

function quadraticRoot(a, b, c) {
  let sqrt = Complex(b * b - 4 * a * c).sqrt()
  let x1 = Complex(-b).add(sqrt).div(2 * a)
  let x2 = Complex(-b).sub(sqrt).div(2 * a)
  return {x1, x2}
}

// quadraticRoot(1, 4, 5) -> -2 ± i

Parser

Any function (see below) as well as the constructor of the Complex class parses its input like this.

You can pass either Objects, Doubles or Strings.

Objects

new Complex({re: real, im: imaginary});
new Complex({arg: angle, abs: radius});
new Complex({phi: angle, r: radius});
new Complex([real, imaginary]); // Vector/Array syntax

If there are other attributes on the passed object, they're not getting preserved and have to be merged manually.

Doubles

new Complex(55.4);

Strings

new Complex("123.45");
new Complex("15+3i");
new Complex("i");

Two arguments

new Complex(3, 2); // 3+2i

Attributes

Every complex number object exposes its real and imaginary part as attribute re and im:

let c = new Complex(3, 2);

console.log("Real part:", c.re); // 3
console.log("Imaginary part:", c.im); // 2

Functions

Complex sign()

Returns the complex sign, defined as the complex number normalized by it's absolute value

Complex add(n)

Adds another complex number

Complex sub(n)

Subtracts another complex number

Complex mul(n)

Multiplies the number with another complex number

Complex div(n)

Divides the number by another complex number

Complex pow(exp)

Returns the number raised to the complex exponent (Note: Complex.ZERO.pow(0) = Complex.ONE by convention)

Complex sqrt()

Returns the complex square root of the number

Complex exp(n)

Returns e^n with complex exponent n.

Complex log()

Returns the natural logarithm (base E) of the actual complex number

Note: The logarithm to a different base can be calculated with z.log().div(Math.log(base)).

double abs()

Calculates the magnitude of the complex number

double arg()

Calculates the angle of the complex number

Complex inverse()

Calculates the multiplicative inverse of the complex number (1 / z)

Complex conjugate()

Calculates the conjugate of the complex number (multiplies the imaginary part with -1)

Complex neg()

Negates the number (multiplies both the real and imaginary part with -1) in order to get the additive inverse

Complex floor([places=0])

Floors the complex number parts towards zero

Complex ceil([places=0])

Ceils the complex number parts off zero

Complex round([places=0])

Rounds the complex number parts

boolean equals(n)

Checks if both numbers are exactly the same, if both numbers are infinite they are considered not equal.

boolean isNaN()

Checks if the given number is not a number

boolean isFinite()

Checks if the given number is finite

Complex clone()

Returns a new Complex instance with the same real and imaginary properties

Array toVector()

Returns a Vector of the actual complex number with two components

String toString()

Returns a string representation of the actual number. As of v1.9.0 the output is a bit more human readable

new Complex(1, 2).toString(); // 1 + 2i
new Complex(0, 1).toString(); // i
new Complex(9, 0).toString(); // 9
new Complex(1, 1).toString(); // 1 + i

double valueOf()

Returns the real part of the number if imaginary part is zero. Otherwise null

Trigonometric functions

The following trigonometric functions are defined on Complex.js:

Trig	Arcus	Hyperbolic	Area-Hyperbolic
sin()	asin()	sinh()	asinh()
cos()	acos()	cosh()	acosh()
tan()	atan()	tanh()	atanh()
cot()	acot()	coth()	acoth()
sec()	asec()	sech()	asech()
csc()	acsc()	csch()	acsch()

Geometric Equivalence

Complex numbers can also be seen as a vector in the 2D space. Here is a simple overview of basic operations and how to implement them with complex.js:

New vector

let v1 = new Complex(1, 0);
let v2 = new Complex(1, 1);

Scale vector

scale(v1, factor):= v1.mul(factor)

Vector norm

norm(v):= v.abs()

Translate vector

translate(v1, v2):= v1.add(v2)

Rotate vector around center

rotate(v, angle):= v.mul({abs: 1, arg: angle})

Rotate vector around a point

rotate(v, p, angle):= v.sub(p).mul({abs: 1, arg: angle}).add(p)

Distance to another vector

distance(v1, v2):= v1.sub(v2).abs()

Constants

Complex.ZERO

A complex zero value (south pole on the Riemann Sphere)

Complex.ONE

A complex one instance

Complex.INFINITY

A complex infinity value (north pole on the Riemann Sphere)

Complex.NAN

A complex NaN value (not on the Riemann Sphere)

Complex.I

An imaginary number i instance

Complex.PI

A complex PI instance

Complex.E

A complex euler number instance

Complex.EPSILON

A small epsilon value used for equals() comparison in order to circumvent double imprecision.

Installation

Installing complex.js is as easy as cloning this repo or use one of the following commands:

bower install complex.js

or

npm install complex.js

Using Complex.js with the browser

<script src="complex.js"></script>
<script>
    console.log(Complex("4+3i"));
</script>

Using Complex.js with require.js

<script src="require.js"></script>
<script>
requirejs(['complex.js'],
function(Complex) {
    console.log(Complex("4+3i"));
});
</script>

Coding Style

As every library I publish, complex.js is also built to be as small as possible after compressing it with Google Closure Compiler in advanced mode. Thus the coding style orientates a little on maxing-out the compression rate. Please make sure you keep this style if you plan to extend the library.

Testing

If you plan to enhance the library, make sure you add test cases and all the previous tests are passing. You can test the library with

npm test

Copyright and licensing

Copyright (c) 2015, Robert Eisele Dual licensed under the MIT or GPL Version 2 licenses.