Struct Complex

Represents a complex number of the form a + bi, where a is the real part and b is the imaginary part. Provides arithmetic operations, polar form conversions, and common complex number properties.

Implements

IEquatable<Complex>

Inherited Members

object.Equals(object, object)

object.GetType()

object.ReferenceEquals(object, object)

Namespace: Scylla.Core.Util.Math

Assembly: ScyllaCore.dll

Syntax

[Serializable]
public struct Complex : IEquatable<Complex>

Remarks

A complex number extends the real number line to a two-dimensional plane. The real part corresponds to the horizontal axis and the imaginary part corresponds to the vertical axis of the complex plane.

This struct is serializable and can be exposed in the Unity Inspector. It is also implicitly convertible to and from float (treating the value as a real number with zero imaginary part) and to and from UnityEngine.Vector2 (treating x as real and y as imaginary).

Equality comparisons use Approximately(float, float, float) to tolerate floating-point rounding errors rather than requiring exact bit-level equality.

Constructors

Complex(float, float)

Initializes a new Complex with the specified real and imaginary parts.

Declaration

public Complex(float real, float imaginary)

Parameters

Type	Name	Description
float	real	The real part of the complex number.
float	imaginary	The imaginary part of the complex number.

Fields

I

The imaginary unit i, defined by the property i^2 = -1. This is the complex number at 90 degrees on the unit circle, i.e., (0, 1) in the complex plane.

Declaration

public static readonly Complex I

Field Value

Type	Description
Complex

Imaginary

The imaginary part of this complex number. Corresponds to the vertical axis of the complex plane.

Declaration

[SerializeField]
public float Imaginary

Field Value

Type	Description
float

One

The complex number 1 + 0i (multiplicative identity). Multiplying any complex number by One leaves it unchanged.

Declaration

public static readonly Complex One

Field Value

Type	Description
Complex

Real

The real part of this complex number. Corresponds to the horizontal axis of the complex plane.

Declaration

[SerializeField]
public float Real

Field Value

Type	Description
float

Zero

The complex number 0 + 0i (additive identity). Adding Zero to any complex number leaves it unchanged.

Declaration

public static readonly Complex Zero

Field Value

Type	Description
Complex

Properties

Argument

Gets the argument (phase angle) of this complex number in radians. This is the angle that the line from the origin to (Real, Imaginary) makes with the positive real axis, computed via atan2(Imaginary, Real).

Declaration

public float Argument { get; }

Property Value

Type	Description
float	A float in the range [-PI, PI]. Returns 0 for real numbers and PI/2 for I.

Conjugate

Gets the complex conjugate of this number. For a complex number a + bi, the conjugate is a - bi. The conjugate reflects the number across the real axis and is used to compute division and to extract real parts from products.

Declaration

public Complex Conjugate { get; }

Property Value

Type	Description
Complex	A new Complex with the same real part and negated imaginary part.

IsPurelyImaginary

Gets a value indicating whether this complex number is purely imaginary (i.e., its real part is negligibly close to zero). A purely imaginary number lies on the vertical axis of the complex plane.

Declaration

public bool IsPurelyImaginary { get; }

Property Value

Type	Description
bool	true if Real is within floating-point tolerance of zero; otherwise false.

IsReal

Gets a value indicating whether this complex number is purely real (i.e., its imaginary part is negligibly close to zero). A real complex number lies on the horizontal axis of the complex plane.

Declaration

public bool IsReal { get; }

Property Value

Type	Description
bool	true if Imaginary is within floating-point tolerance of zero; otherwise false.

Modulus

Gets the modulus (magnitude) of this complex number. This is the Euclidean distance from the origin to the point (Real, Imaginary) on the complex plane, computed as sqrt(Real^2 + Imaginary^2).

Declaration

public float Modulus { get; }

Property Value

Type	Description
float	A non-negative float representing the distance from the origin. Returns 0 for Zero.

See Also

SqrModulus

SqrModulus

Gets the squared modulus (magnitude squared) of this complex number, computed as Real^2 + Imaginary^2. Avoids the square root of Modulus and is therefore more efficient when only comparing relative magnitudes.

Declaration

public float SqrModulus { get; }

Property Value

Type	Description
float	A non-negative float. Equal to the product of this number and its Conjugate's real part.

See Also

Modulus

Methods

Abs(Complex)

Returns the absolute value (modulus) of a complex number. This is a static functional equivalent of the Modulus property.

Declaration

public static float Abs(Complex z)

Parameters

Type	Name	Description
Complex	z	The complex number whose modulus is to be computed.

Returns

Type	Description
float	The Euclidean distance from the origin to z on the complex plane.

See Also

Modulus

Cbrt(Complex)

Returns the principal cube root of a complex number. The result w satisfies w * w * w == z on the principal branch (argument in (-PI/3, PI/3]).

Declaration

public static Complex Cbrt(Complex z)

Parameters

Type	Name	Description
Complex	z	The complex number whose cube root is to be computed.

Returns

Type	Description
Complex	A Complex whose modulus is |z|^(1/3) and whose argument is arg(z) / 3.

See Also

Sqrt(Complex)

Cis(float)

Returns the complex number cis(theta), defined as cos(theta) + isin(theta). This is equivalent to e^(itheta) by Euler's formula and always lies on the unit circle.

Declaration

public static Complex Cis(float theta)

Parameters

Type	Name	Description
float	theta	The angle in radians, measured from the positive real axis.

Returns

Type	Description
Complex	A Complex on the unit circle at angle theta, with Modulus equal to 1.

See Also

FromPolar(float, float)

Equals(Complex)

Determines whether this Complex is equal to another Complex. Equality is checked using Approximately(float, float, float) to tolerate floating-point rounding errors.

Declaration

public bool Equals(Complex other)

Parameters

Type	Name	Description
Complex	other	The Complex to compare with this instance.

Returns

Type	Description
bool	true if both the Real and Imaginary parts of other are approximately equal to those of this instance; otherwise false.

Equals(object)

Determines whether this Complex is equal to another object.

Declaration

public override bool Equals(object obj)

Parameters

Type	Name	Description
object	obj	The object to compare with this instance.

Returns

Type	Description
bool	true if obj is a Complex with approximately equal real and imaginary parts; otherwise false.

Overrides

ValueType.Equals(object)

FromPolar(float, float)

Creates a complex number from polar coordinates using Euler's formula: r * e^(itheta) = r * (cos(theta) + isin(theta)).

Declaration

public static Complex FromPolar(float r, float theta)

Parameters

Type	Name	Description
float	r	The modulus (distance from the origin). A value of 1 places the result on the unit circle.
float	theta	The argument (angle) in radians, measured from the positive real axis.

Returns

Type	Description
Complex	A Complex whose Real part is r * cos(theta) and whose Imaginary part is r * sin(theta).

See Also

Cis(float)

GetHashCode()

Returns a hash code for this Complex. The hash is computed from both the Real and Imaginary parts using a combined hash strategy.

Declaration

public override int GetHashCode()

Returns

Type	Description
int	An int hash code suitable for use in hash-based collections.

Overrides

ValueType.GetHashCode()

Im(Complex)

Returns the imaginary part of a complex number. This is a static functional equivalent of accessing the Imaginary field directly.

Declaration

public static float Im(Complex z)

Parameters

Type	Name	Description
Complex	z	The complex number whose imaginary part is to be extracted.

Returns

Type	Description
float	The value of z.Imaginary.

Pow(Complex, float)

Returns z raised to the real power p using polar form: |z|^p * e^(i * arg(z) * p).

Declaration

public static Complex Pow(Complex z, float p)

Parameters

Type	Name	Description
Complex	z	The complex base.
float	p	The real exponent. Fractional and negative values are supported.

Returns

Type	Description
Complex	A Complex with modulus |z|^p and argument arg(z) * p.

Re(Complex)

Returns the real part of a complex number. This is a static functional equivalent of accessing the Real field directly.

Declaration

public static float Re(Complex z)

Parameters

Type	Name	Description
Complex	z	The complex number whose real part is to be extracted.

Returns

Type	Description
float	The value of z.Real.

RootOfUnity(int, int)

Returns the k-th n-th root of unity, i.e., the k-th solution to the equation z^n = 1. All roots of unity lie on the unit circle and are evenly spaced by an angle of 2*PI/n.

Declaration

public static Complex RootOfUnity(int n, int k = 1)

Parameters

Type	Name	Description
int	n	The degree of the root. Must be a positive integer greater than zero.
int	k	The index of the root to return, in the range [0, n-1]. Defaults to 1, which gives the principal primitive root at angle 2*PI/n. Passing 0 always returns One.

Returns

Type	Description
Complex	The complex number e^(2PIki/n) = cos(2PIk/n) + isin(2PIk/n).

Remarks

This method is used internally by GetRoots() when computing the three real roots of a depressed cubic via Cardano's formula.

Sqrt(Complex)

Returns the principal square root of a complex number. The result is chosen such that its argument lies in the range (-PI/2, PI/2] (i.e., the principal branch). The result w satisfies w * w == z.

Declaration

public static Complex Sqrt(Complex z)

Parameters

Type	Name	Description
Complex	z	The complex number whose square root is to be computed.

Returns

Type	Description
Complex	A Complex whose modulus is |z|^0.5 and whose argument is arg(z) / 2.

See Also

Sqrt(float)

Cbrt(Complex)

Sqrt(float)

Returns the square root of a real number as a complex result, allowing negative inputs to produce a purely imaginary result rather than NaN.

Declaration

public static Complex Sqrt(float f)

Parameters

Type	Name	Description
float	f	The real number to take the square root of. May be negative.

Returns

Type	Description
Complex	If f is non-negative, returns a real Complex with Real equal to sqrt(f) and zero Imaginary. If f is negative, returns a purely imaginary Complex with zero Real and Imaginary equal to sqrt(-f).

See Also

Sqrt(Complex)

ToString()

Returns a string representation of this complex number in standard mathematical notation. Special cases are handled for purely real numbers, purely imaginary numbers, and unit imaginary parts.

Declaration

public override string ToString()

Returns

Type	Description
string	If purely real: the real part formatted as an invariant-culture float (e.g., "3.5"). If purely imaginary with magnitude 1: "i" or "-i". If purely imaginary: the imaginary part followed by "i" (e.g., "2.5i"). Otherwise: standard notation such as "3 + 2i" or "1 - 4i".

Overrides

ValueType.ToString()

Operators

operator +(Complex, Complex)

Adds two complex numbers by summing their real and imaginary parts independently.

Declaration

public static Complex operator +(Complex z, Complex w)

Parameters

Type	Name	Description
Complex	z	The left-hand addend.
Complex	w	The right-hand addend.

Returns

Type	Description
Complex	A Complex equal to (z.Real + w.Real) + (z.Imaginary + w.Imaginary)i.

operator /(Complex, Complex)

Divides one complex number by another using conjugate multiplication: z / w = (z * conjugate(w)) / |w|^2. This avoids complex division by converting the denominator to a real number.

Declaration

public static Complex operator /(Complex z, Complex w)

Parameters

Type	Name	Description
Complex	z	The dividend.
Complex	w	The divisor. Must not be the zero complex number.

Returns

Type	Description
Complex	A Complex equal to z divided by w.

Exceptions

Type	Condition
DivideByZeroException	Thrown when w has a SqrModulus that is negligibly close to zero (i.e., when dividing by Zero).

operator ==(Complex, Complex)

Returns true if two Complex values are approximately equal, comparing both their real and imaginary parts using Approximately(float, float, float).

Declaration

public static bool operator ==(Complex z, Complex w)

Parameters

Type	Name	Description
Complex	z	The left-hand operand.
Complex	w	The right-hand operand.

Returns

Type	Description
bool	true if z and w are approximately equal; otherwise false.

implicit operator Vector2(Complex)

Implicitly converts a Complex to a UnityEngine.Vector2, treating the real part as x and the imaginary part as y. This allows complex numbers to be used directly as 2D vectors.

Declaration

public static implicit operator Vector2(Complex z)

Parameters

Type	Name	Description
Complex	z	The complex number to convert.

Returns

Type	Description
Vector2	A UnityEngine.Vector2 with x equal to z.Real and y equal to z.Imaginary.

implicit operator Complex(float)

Implicitly converts a float real number to a Complex with a zero imaginary part, placing it on the real axis of the complex plane.

Declaration

public static implicit operator Complex(float real)

Parameters

Type	Name	Description
float	real	The real value to convert.

Returns

Type	Description
Complex	A Complex with Real set to real and Imaginary set to 0.

implicit operator Complex(Vector2)

Implicitly converts a UnityEngine.Vector2 to a Complex, treating x as the real part and y as the imaginary part. This allows 2D vectors to be used directly in complex arithmetic.

Declaration

public static implicit operator Complex(Vector2 v)

Parameters

Type	Name	Description
Vector2	v	The vector to convert.

Returns

Type	Description
Complex	A Complex with Real equal to v.x and Imaginary equal to v.y.

operator !=(Complex, Complex)

Returns true if two Complex values are not approximately equal.

Declaration

public static bool operator !=(Complex z, Complex w)

Parameters

Type	Name	Description
Complex	z	The left-hand operand.
Complex	w	The right-hand operand.

Returns

Type	Description
bool	true if z and w are not approximately equal; otherwise false.

operator *(Complex, Complex)

Multiplies two complex numbers using the standard algebraic formula: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.

Declaration

public static Complex operator *(Complex z, Complex w)

Parameters

Type	Name	Description
Complex	z	The left-hand factor.
Complex	w	The right-hand factor.

Returns

Type	Description
Complex	A Complex equal to (z.Realw.Real - z.Imaginaryw.Imaginary) + (z.Realw.Imaginary + z.Imaginaryw.Real)i.

operator -(Complex, Complex)

Subtracts one complex number from another by subtracting their real and imaginary parts independently.

Declaration

public static Complex operator -(Complex z, Complex w)

Parameters

Type	Name	Description
Complex	z	The minuend.
Complex	w	The subtrahend.

Returns

Type	Description
Complex	A Complex equal to (z.Real - w.Real) + (z.Imaginary - w.Imaginary)i.

operator -(Complex)

Unary negation operator. Returns the additive inverse of the complex number, negating both the real and imaginary parts.

Declaration

public static Complex operator -(Complex z)

Parameters

Type	Name	Description
Complex	z	The operand to negate.

Returns

Type	Description
Complex	A Complex equal to -Real - Imaginary*i.

operator +(Complex)

Unary plus operator. Returns the complex number unchanged.

Declaration

public static Complex operator +(Complex z)

Parameters

Type	Name	Description
Complex	z	The operand.

Returns

Type	Description
Complex	The same Complex value as z.

Implements

IEquatable<T>

See Also

Polynomial

Monomial