science

Complex numbers in polar coordinates

Sometimes complex number calculations simplifies significantly when performed in polar coordinates form i.e. expressed as radius (magnitude) and phase angle.

You can swap easily using Complex::toPolar() and ComplexPolar::toRectangular() methods.

Basic usage

use irrevion\science\Math\Math;
use irrevion\science\Math\Entities\{Scalar, Imaginary, Complex, ComplexPolar};

$x = new ComplexPolar(4.27, Math::PI/4);
print("ComplexPolar(4.27, Math::PI/4) to string is {$x}\n"); // outputs ComplexPolar(4.27, Math::PI/4) to string is [4.27, φ 0.25π RAD]

$y = new ComplexPolar(3, Math::PI/6);

$sum = $x->add($y); // result is [6.93, φ 0.35π RAD]
$diff = $x->subtract($y); // result is [1.27, φ 0.15π RAD]
$prod = $x->multiply($y); // result is [12.81, φ 0.75π RAD]
$quotient = $x->divide($y); // result is [1.42, φ 0.083π RAD]
$reciprocal = $x->reciprocal(); // result is [0.234, φ -0.25π RAD]
$conjugate = $x->conjugate(); // result is [4.27, φ -0.25π RAD]
$magnitude = $x->abs(); // result is 4.27
$phase = $x->arg(); // result is 0.25π RAD
$pow = $x->pow(2); // result is [18.23, φ 0.5π RAD]
$exp = $x->exp(); // result is [71.8, φ 0.25π RAD]
$ln = $x->ln(); // result is [1.45, φ 0.25π RAD]

$is_empty = (new ComplexPolar(0, 0))->empty(); // true
$is_equal = $x->isEqual($x); // true
$is_near = $x->isNear(new ComplexPolar(4.27, Math::PI/4 + 0.0001)); // true

Constructor

You can create an instance of a ComplexPolar number in different ways. This is something already familiar:

$x = new ComplexPolar(4.27, Math::PI/4);
$y = new ComplexPolar(3, Math::PI/6);

You can pass an instance of Complex:

$c = new Complex(4, 0.92);
$cp = new ComplexPolar($c);

And Scalar is acceptable as well (phase will be taken as zero):

$s = new Scalar(-12);
$cps = new ComplexPolar($s);

Arrays passed to constructor are not refused at all:

$cpa = new ComplexPolar([
	'radius' => 4,
	'phase' => 0.92,
]);

You can also create a copy of another ComplexPolar number:

$cp1 = new ComplexPolar(2.5, Math::PI/3);
$cp2 = new ComplexPolar($cp1);

Addition / Subtraction

As denoted above, there are methods for addition and subtraction of complex polar numbers:

$x = new ComplexPolar(4.27, Math::PI/4);
$y = new ComplexPolar(3, Math::PI/6);
$sum = $x->add($y); // result is [6.93, φ 0.35π RAD]
$diff = $x->subtract($y); // result is [1.27, φ 0.15π RAD]

Multiplication / Division

Complex polar number multiplication, division, obtaining reciprocal and conjugate methods are pretty straightforward:

$prod = $x->multiply($y); // result is [12.81, φ 0.75π RAD]
$quotient = $x->divide($y); // result is [1.42, φ 0.083π RAD]
$reciprocal = $x->reciprocal(); // result is [0.234, φ -0.25π RAD]
$conjugate = $x->conjugate(); // result is [4.27, φ -0.25π RAD]

Other methods

Natural logarithm:

print (new ComplexPolar(2, Math::PI/3))->ln(); // outputs [0.693, φ 0.333π RAD]

Exponent:

print (new ComplexPolar(1, Math::PI/4))->exp(); // outputs [2.718, φ 0.25π RAD]