orbital.math
Interface Complex

All Superinterfaces:

Arithmetic, Normed, Scalar

All Known Subinterfaces:

Integer, Rational, Real

public interface Complex
extends Scalar

Representation of a complex number a + i*b∈C.

Complex numbers are a field that is a superset of R⊂C. They are isomorph to a special polynomial ring, C ≅ R[i] / (i² = -1) = R[X] / (X²+1).

☡ Since complex numbers are a field, they are also an Euclidean ring. However, because fields are (in a way) "degenerate" Euclidean rings, they do not extend the Euclidean interface. The Euclidean quotient and remainder operations would not perform very interesting calculations, and the Euclidean quotient would fully coincide with ordinary division in a field.

☡ Complex numbers are not like real numbers. They are not ordered.

Author:

André Platzer

See Also:

ValueFactory.complex(Real, Real), ValueFactory.complex(double, double), ValueFactory.cartesian(Real, Real), ValueFactory.cartesian(double, double), ValueFactory.polar(Real, Real), ValueFactory.polar(double, double)

Invariants:

(¬super ∧ ¬(this is Comparable)) ∨ □abnormal(¬ Comparable)

Stereotype:

data-type

Field Summary

static Predicate	hasType Checks whether the given number is a proper complex, i.e.
static Predicate	isa Checks whether the given number is in the set of complex numbers.

Fields inherited from interface orbital.math.Arithmetic

numerical

Method Summary

Complex	add(Complex b) adds two Complexes returning a third as a result
Real	arg() Returns the principal angle (argument) component of a polar complex.
Complex	conjugate() Returns the complex conjugated z = z* = z'.
Complex	divide(Complex b) divides two complex numbers.
boolean	equals(java.lang.Object o) Compares two complex numbers for equality.
int	hashCode() Returns a hash code value for the object.
Real	im() Get the imaginar component.
boolean	isInfinite() Whether this complex number is infinite.
boolean	isNaN() Whether this complex number is NaN.
Complex	multiply(Complex b) multiplies two Complexes returning a third as a result
Real	norm() Returns the absolute |z|.
Complex	power(Complex x) power of complex numbers.
Real	re() Get the real component.
Complex	subtract(Complex b) subtracts two Complexes returning a third as a result

Methods inherited from interface orbital.math.Arithmetic

add, divide, equals, inverse, isOne, isZero, minus, multiply, one, power, scale, subtract, toString, valueFactory, zero

Field Detail

isa

static final Predicate isa

Checks whether the given number is in the set of complex numbers. return whether v is complex, real, rational or an integer.

hasType

static final Predicate hasType

Checks whether the given number is a proper complex, i.e. it is in the set of complexes and not a subset. return whether v∈C\R is complex, but not real.

Method Detail

equals

boolean equals(java.lang.Object o)

Compares two complex numbers for equality.

Two complex numbers z, z' ∈ C are equal iff re z = re z' ∧ im z = im z'. This is a component-wise equality.

Overrides:

equals in class java.lang.Object

Preconditions:

true

Postconditions:

RES ⇔ Complex.isa(o) ∧ re() = o.re() ∧ im() = o.im()

hashCode

int hashCode()

Returns a hash code value for the object.

Overrides:

hashCode in class java.lang.Object

Preconditions:

true

Postconditions:

RES == re().hashCode() ^ im().hashCode()

re

Real re()

Get the real component.

Returns:

ℜ(z) = re z = re (a + ib) = a = (z+z) / 2

See Also:

im()

im

Real im()

Get the imaginar component.

Returns:

ℑ(z) = im z = im (a + ib) = b = (z-z) / (2i)

See Also:

re()

norm

Real norm()

Returns the absolute |z|.

It is |z|² = z*z = re(z)²+im(z)². and |x*y| = |x|*|y|, |z^-1| = |z|^-1.

Also called length, magnitude or modulus.

Specified by:

norm in interface Normed

Returns:

the absolute |z| = √re(z)²+im(z)².

See Also:

arg()

arg

Real arg()

Returns the principal angle (argument) component of a polar complex. But adding 2kπ to the principle angle will be an angle as well.

Returns:

the angle φ of r*e^iφ. The angle ∠ φ in radians is measured counter-clockwise from the real axis. Value will be in range [-π,π] and is not yet in the correct sector!

See Also:

norm()

conjugate

Complex conjugate()

Returns the complex conjugated z = z^* = z'.

Conjugation is an involutive field-automorphism that is identical on R.

Returns:

the complex number a - i*b.

isInfinite

boolean isInfinite()

Whether this complex number is infinite.

Returns:

whether the real or imaginary part is infinite.

See Also:

Double.isInfinite(double)

isNaN

boolean isNaN()

Whether this complex number is NaN.

Returns:

whether the real or imaginary part is NaN.

See Also:

Double.isNaN(double)

add

Complex add(Complex b)

adds two Complexes returning a third as a result

subtract

Complex subtract(Complex b)

subtracts two Complexes returning a third as a result

multiply

Complex multiply(Complex b)

multiplies two Complexes returning a third as a result

divide

Complex divide(Complex b)

divides two complex numbers.

power

Complex power(Complex x)

power of complex numbers.