Complex (Apache Commons Numbers Complex 1.0-SNAPSHOT API)

java.lang.Object
- org.apache.commons.numbers.complex.Complex

All Implemented Interfaces:

Serializable
```
public final class Complex
extends Object
implements Serializable
```
Representation of a Complex number, i.e. a number which has both a real and imaginary part.
Implementations of arithmetic operations handle NaN and infinite values according to the rules for Double, i.e. equals(java.lang.Object) is an equivalence relation for all instances that have a NaN in either real or imaginary part, e.g. the following are considered equal:
- 1 + NaNi
- NaN + i
- NaN + NaNi
Note that this contradicts the IEEE-754 standard for floating point numbers (according to which the test x == x must fail if x is NaN). The method equals for primitive double in class Precision conforms with IEEE-754 while this class conforms with the standard behavior for Java object types.
See Also:

Serialized Form

Field Summary

Fields
Modifier and Type	Field and Description
`static Complex`	`I` The square root of -1, a.k.a.
`static Complex`	`INF` A complex number representing "+INF + INF i"
`static Complex`	`ONE` A complex number representing one.
`static Complex`	`ZERO` A complex number representing zero.

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`abs()` Return the absolute value of this complex number.
`Complex`	`acos()` Compute the inverse cosine of this complex number.
`Complex`	`acosh()` Compute the inverse hyperbolic cosine of this complex number.
`Complex`	`add(Complex addend)` Returns a `Complex` whose value is `(this + addend)`.
`Complex`	`add(double addend)` Returns a `Complex` whose value is `(this + addend)`, with `addend` interpreted as a real number.
`double`	`arg()` Compute the argument of this complex number.
`Complex`	`asin()` Compute the inverse sine of this complex number.
`Complex`	`asinh()` Compute the inverse hyperbolic sine of this complex number.
`Complex`	`atan()` Compute the inverse tangent of this complex number.
`Complex`	`atanh()` Compute the inverse hyperbolic tangent of this complex number.
`Complex`	`conj()` Returns the conjugate of this complex number.
`Complex`	`conjugate()` Returns the conjugate of this complex number.
`Complex`	`cos()` Compute the cosine of this complex number.
`Complex`	`cosh()` Compute the hyperbolic cosine of this complex number.
`Complex`	`divide(Complex divisor)` Returns a `Complex` whose value is `(this / divisor)`.
`Complex`	`divide(double divisor)` Returns a `Complex` whose value is `(this / divisor)`, with `divisor` interpreted as a real number.
`static boolean`	`equals(Complex x, Complex y)` Returns `true` iff the values are equal as defined by `equals(x, y, 1)`.
`static boolean`	`equals(Complex x, Complex y, double eps)` Returns `true` if, both for the real part and for the imaginary part, there is no double value strictly between the arguments or the difference between them is within the range of allowed error (inclusive).
`static boolean`	`equals(Complex x, Complex y, int maxUlps)` Test for the floating-point equality between Complex objects.
`boolean`	`equals(Object other)` Test for equality with another object.
`static boolean`	`equalsWithRelativeTolerance(Complex x, Complex y, double eps)` Returns `true` if, both for the real part and for the imaginary part, there is no double value strictly between the arguments or the relative difference between them is smaller or equal to the given tolerance.
`Complex`	`exp()` Compute the exponential function of this complex number.
`double`	`getArgument()` Compute the argument of this complex number.
`double`	`getImaginary()` Access the imaginary part.
`double`	`getReal()` Access the real part.
`int`	`hashCode()` Get a hash code for the complex number.
`double`	`imag()` Access the imaginary part (C++ grammar)
`boolean`	`isInfinite()` Returns true if either real or imaginary component of the Complex is Infinite
`boolean`	`isNaN()` Returns true if either real or imaginary component of the Complex is NaN
`Complex`	`log()` Compute the natural logarithm of this complex number.
`Complex`	`log10()` Compute the base 10 or common logarithm of this complex number.
`Complex`	`multiply(Complex factor)` Returns a `Complex` whose value is `this * factor`.
`Complex`	`multiply(double factor)` Returns a `Complex` whose value is `this * factor`, with `factor` interpreted as a real number.
`Complex`	`multiply(int factor)` Returns a `Complex` whose value is `this * factor`, with `factor` interpreted as a integer number.
`Complex`	`negate()` Returns a `Complex` whose value is `(-this)`.
`List<Complex>`	`nthRoot(int n)` Computes the n-th roots of this complex number.
`static Complex`	`ofCartesian(double real, double imaginary)` Create a complex number given the real and imaginary parts.
`static Complex`	`ofCis(double x)` For a real constructor argument x, returns a new Complex object c where `c = cos(x) + i sin (x)`
`static Complex`	`ofPolar(double r, double theta)` Creates a Complex from its polar representation.
`static Complex`	`ofReal(double real)` Create a complex number given the real part.
`static Complex`	`parse(String s)` Parses a string that would be produced by `toString()` and instantiates the corresponding object.
`Complex`	`pow(Complex x)` Returns of value of this complex number raised to the power of `x`.
`Complex`	`pow(double x)` Returns of value of this complex number raised to the power of `x`.
`Complex`	`proj()` Returns projection of this complex number onto the Riemann sphere, i.e.
`double`	`real()` Access the real part (C++ grammar)
`Complex`	`reciprocal()` Returns the multiplicative inverse of this instance.
`Complex`	`sin()` Compute the sine of this complex number.
`Complex`	`sinh()` Compute the hyperbolic sine of this complex number.
`Complex`	`sqrt()` Compute the square root of this complex number.
`Complex`	`square()` Compute the square of this complex number.
`Complex`	`subtract(Complex subtrahend)` Returns a `Complex` whose value is `(this - subtrahend)`.
`Complex`	`subtract(double subtrahend)` Returns a `Complex` whose value is `(this - subtrahend)`.
`Complex`	`tan()` Compute the tangent of this complex number.
`Complex`	`tanh()` Compute the hyperbolic tangent of this complex number.
`String`	`toString()`

Methods inherited from class java.lang.Object
clone, finalize, getClass, notify, notifyAll, wait, wait, wait

- Field Detail
  - I
```
public static final Complex I
```
    The square root of -1, a.k.a. "i".
  - INF
```
public static final Complex INF
```
    A complex number representing "+INF + INF i"
  - ONE
```
public static final Complex ONE
```
    A complex number representing one.
  - ZERO
```
public static final Complex ZERO
```
    A complex number representing zero.
- Method Detail
  - ofCartesian
```
public static Complex ofCartesian(double real,
                                  double imaginary)
```
    Create a complex number given the real and imaginary parts.
    
    Parameters:
    
    real - Real part.
    
    imaginary - Imaginary part.
    
    Returns:
    
    Complex object
  - ofReal
```
public static Complex ofReal(double real)
```
    Create a complex number given the real part.
    
    Parameters:
    
    real - Real part.
    
    Returns:
    
    Complex object
  - ofPolar
```
public static Complex ofPolar(double r,
                              double theta)
```
    Creates a Complex from its polar representation. If r is infinite and theta is finite, infinite or NaN values may be returned in parts of the result, following the rules for double arithmetic.
```
 Examples:
 
 polar2Complex(INFINITY, \(\pi\)) = INFINITY + INFINITY i
 polar2Complex(INFINITY, 0) = INFINITY + NaN i
 polar2Complex(INFINITY, \(-\frac{\pi}{4}\)) = INFINITY - INFINITY i
 polar2Complex(INFINITY, \(5\frac{\pi}{4}\)) = -INFINITY - INFINITY i 
 
```
    Parameters:
    
    r - the modulus of the complex number to create
    
    theta - the argument of the complex number to create
    
    Returns:
    
    Complex
  - ofCis
```
public static Complex ofCis(double x)
```
    For a real constructor argument x, returns a new Complex object c where c = cos(x) + i sin (x)
    
    Parameters:
    
    x - double to build the cis number
    
    Returns:
    
    Complex
  - parse
```
public static Complex parse(String s)
```
    Parses a string that would be produced by toString() and instantiates the corresponding object.
    
    Parameters:
    
    s - String representation.
    
    Returns:
    
    an instance.
    
    Throws:
    
    IllegalArgumentException - if the string does not conform to the specification.
  - isNaN
```
public boolean isNaN()
```
    Returns true if either real or imaginary component of the Complex is NaN
    
    Returns:
    
    boolean
  - isInfinite
```
public boolean isInfinite()
```
    Returns true if either real or imaginary component of the Complex is Infinite
    
    Returns:
    
    boolean
  - proj
```
public Complex proj()
```
    Returns projection of this complex number onto the Riemann sphere, i.e. all infinities (including those with an NaN component) project onto real infinity, as described in the IEEE and ISO C standards.
    
    Returns:
    
    Complex projected onto the Riemann sphere.
  - abs
```
public double abs()
```
    Return the absolute value of this complex number. This code follows the ISO C Standard, Annex G, in calculating the returned value (i.e. the hypot(x,y) method) and in handling of NaNs.
    
    Returns:
    
    the absolute value.
  - add
```
public Complex add(Complex addend)
```
    Returns a Complex whose value is (this + addend). Uses the definitional formula
    (a + bi) + (c + di) = (a+c) + (b+d)i
    
    Parameters:
    
    addend - Value to be added to this Complex.
    
    Returns:
    
    this + addend.
  - add
```
public Complex add(double addend)
```
    Returns a Complex whose value is (this + addend), with addend interpreted as a real number.
    
    Parameters:
    
    addend - Value to be added to this Complex.
    
    Returns:
    
    this + addend.
    
    See Also:
    
    add(Complex)
  - conjugate
```
public Complex conjugate()
```
    Returns the conjugate of this complex number. The conjugate of a + bi is a - bi.
    
    Returns:
    
    the conjugate of this complex object.
  - conj
```
public Complex conj()
```
    Returns the conjugate of this complex number. C++11 grammar.
    
    Returns:
    
    the conjugate of this complex object.
  - divide
```
public Complex divide(Complex divisor)
```
    Returns a Complex whose value is (this / divisor). Implements the definitional formula
```
  
    a + bi          ac + bd + (bc - ad)i
    ----------- = -------------------------
    c + di         c² + d²
  
 
```
    Recalculates to recover infinities as specified in C.99 standard G.5.1. Method is fully in accordance with C++11 standards for complex numbers.
    Parameters:
    
    divisor - Value by which this Complex is to be divided.
    
    Returns:
    
    this / divisor.
  - divide
```
public Complex divide(double divisor)
```
    Returns a Complex whose value is (this / divisor), with divisor interpreted as a real number.
    
    Parameters:
    
    divisor - Value by which this Complex is to be divided.
    
    Returns:
    
    this / divisor.
    
    See Also:
    
    divide(Complex)
  - reciprocal
```
public Complex reciprocal()
```
    Returns the multiplicative inverse of this instance.
    
    Returns:
    
    1 / this.
    
    See Also:
    
    divide(Complex)
  - equals
```
public boolean equals(Object other)
```
    Test for equality with another object. If both the real and imaginary parts of two complex numbers are exactly the same, and neither is Double.NaN, the two Complex objects are considered to be equal. The behavior is the same as for JDK's Double:
    - All NaN values are considered to be equal, i.e, if either (or both) real and imaginary parts of the complex number are equal to Double.NaN, the complex number is equal to NaN.
    - Instances constructed with different representations of zero (i.e. either "0" or "-0") are not considered to be equal.
    Overrides:
    
    equals in class Object
    
    Parameters:
    
    other - Object to test for equality with this instance.
    
    Returns:
    
    true if the objects are equal, false if object is null, not an instance of Complex, or not equal to this instance.
  - equals
```
public static boolean equals(Complex x,
                             Complex y,
                             int maxUlps)
```
    Test for the floating-point equality between Complex objects. It returns true if both arguments are equal or within the range of allowed error (inclusive).
    
    Parameters:
    
    x - First value (cannot be null).
    
    y - Second value (cannot be null).
    
    maxUlps - (maxUlps - 1) is the number of floating point values between the real (resp. imaginary) parts of x and y.
    
    Returns:
    
    true if there are fewer than maxUlps floating point values between the real (resp. imaginary) parts of x and y.
    
    See Also:
    
    Precision.equals(double,double,int)
  - equals
```
public static boolean equals(Complex x,
                             Complex y)
```
    Returns true iff the values are equal as defined by equals(x, y, 1).
    
    Parameters:
    
    x - First value (cannot be null).
    
    y - Second value (cannot be null).
    
    Returns:
    
    true if the values are equal.
  - equals
```
public static boolean equals(Complex x,
                             Complex y,
                             double eps)
```
    Returns true if, both for the real part and for the imaginary part, there is no double value strictly between the arguments or the difference between them is within the range of allowed error (inclusive). Returns false if either of the arguments is NaN.
    
    Parameters:
    
    x - First value (cannot be null).
    
    y - Second value (cannot be null).
    
    eps - Amount of allowed absolute error.
    
    Returns:
    
    true if the values are two adjacent floating point numbers or they are within range of each other.
    
    See Also:
    
    Precision.equals(double,double,double)
  - equalsWithRelativeTolerance
```
public static boolean equalsWithRelativeTolerance(Complex x,
                                                  Complex y,
                                                  double eps)
```
    Returns true if, both for the real part and for the imaginary part, there is no double value strictly between the arguments or the relative difference between them is smaller or equal to the given tolerance. Returns false if either of the arguments is NaN.
    
    Parameters:
    
    x - First value (cannot be null).
    
    y - Second value (cannot be null).
    
    eps - Amount of allowed relative error.
    
    Returns:
    
    true if the values are two adjacent floating point numbers or they are within range of each other.
    
    See Also:
    
    Precision.equalsWithRelativeTolerance(double,double,double)
  - hashCode
```
public int hashCode()
```
    Get a hash code for the complex number. Any Double.NaN value in real or imaginary part produces the same hash code 7.
    
    Overrides:
    
    hashCode in class Object
    
    Returns:
    
    a hash code value for this object.
  - getImaginary
```
public double getImaginary()
```
    Access the imaginary part.
    
    Returns:
    
    the imaginary part.
  - imag
```
public double imag()
```
    Access the imaginary part (C++ grammar)
    
    Returns:
    
    the imaginary part.
  - getReal
```
public double getReal()
```
    Access the real part.
    
    Returns:
    
    the real part.
  - real
```
public double real()
```
    Access the real part (C++ grammar)
    
    Returns:
    
    the real part.
  - multiply
```
public Complex multiply(Complex factor)
```
    Returns a Complex whose value is this * factor. Implements the definitional formula: (a + bi)(c + di) = (ac - bd) + (ad + bc)i Recalculates to recover infinities as specified in C.99 standard G.5.1. Method is fully in accordance with C++11 standards for complex numbers.
    
    Parameters:
    
    factor - value to be multiplied by this Complex.
    
    Returns:
    
    this * factor.
  - multiply
```
public Complex multiply(int factor)
```
    Returns a Complex whose value is this * factor, with factor interpreted as a integer number.
    
    Parameters:
    
    factor - value to be multiplied by this Complex.
    
    Returns:
    
    this * factor.
    
    See Also:
    
    multiply(Complex)
  - multiply
```
public Complex multiply(double factor)
```
    Returns a Complex whose value is this * factor, with factor interpreted as a real number.
    
    Parameters:
    
    factor - value to be multiplied by this Complex.
    
    Returns:
    
    this * factor.
    
    See Also:
    
    multiply(Complex)
  - negate
```
public Complex negate()
```
    Returns a Complex whose value is (-this).
    
    Returns:
    
    -this.
  - subtract
```
public Complex subtract(Complex subtrahend)
```
    Returns a Complex whose value is (this - subtrahend). Uses the definitional formula
    (a + bi) - (c + di) = (a-c) + (b-d)i
    
    Parameters:
    
    subtrahend - value to be subtracted from this Complex.
    
    Returns:
    
    this - subtrahend.
  - subtract
```
public Complex subtract(double subtrahend)
```
    Returns a Complex whose value is (this - subtrahend).
    
    Parameters:
    
    subtrahend - value to be subtracted from this Complex.
    
    Returns:
    
    this - subtrahend.
    
    See Also:
    
    subtract(Complex)
  - acos
```
public Complex acos()
```
    Compute the inverse cosine of this complex number. Implements the formula:
    acos(z) = -i (log(z + i (sqrt(1 - z<sup>2</sup>))))
    
    Returns:
    
    the inverse cosine of this complex number.
  - asin
```
public Complex asin()
```
    Compute the inverse sine of this complex number.
    asin(z) = -i (log(sqrt(1 - z<sup>2</sup>) + iz))
    
    Returns:
    
    the inverse sine of this complex number
  - atan
```
public Complex atan()
```
    Compute the inverse tangent of this complex number. Implements the formula:
    atan(z) = (i/2) log((i + z)/(i - z))
    
    Returns:
    
    the inverse tangent of this complex number
  - asinh
```
public Complex asinh()
```
    Compute the inverse hyperbolic sine of this complex number. Implements the formula:
    asinh(z) = log(z+sqrt(z^2+1))
    
    Returns:
    
    the inverse hyperbolic cosine of this complex number
  - atanh
```
public Complex atanh()
```
    Compute the inverse hyperbolic tangent of this complex number. Implements the formula:
    atanh(z) = log((1+z)/(1-z))/2
    
    Returns:
    
    the inverse hyperbolic cosine of this complex number
  - acosh
```
public Complex acosh()
```
    Compute the inverse hyperbolic cosine of this complex number. Implements the formula:
    acosh(z) = log(z+sqrt(z^2-1))
    
    Returns:
    
    the inverse hyperbolic cosine of this complex number
  - square
```
public Complex square()
```
    Compute the square of this complex number.
    
    Returns:
    
    square of this complex number
  - cos
```
public Complex cos()
```
    Compute the cosine of this complex number. Implements the formula:
    cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b)i
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), Math.cosh(double) and Math.sinh(double).
    
    Returns:
    
    the cosine of this complex number.
  - cosh
```
public Complex cosh()
```
    Compute the hyperbolic cosine of this complex number. Implements the formula:
```
  
   cosh(a + bi) = cosh(a)cos(b) + sinh(a)sin(b)i
  
 
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), Math.cosh(double) and Math.sinh(double).
    Returns:
    
    the hyperbolic cosine of this complex number.
  - exp
```
public Complex exp()
```
    Compute the exponential function of this complex number. Implements the formula:
```
  
   exp(a + bi) = exp(a)cos(b) + exp(a)sin(b)i
  
 
```
    where the (real) functions on the right-hand side are Math.exp(double), Math.cos(double), and Math.sin(double).
    Returns:
    
    e^this.
  - log
```
public Complex log()
```
    Compute the natural logarithm of this complex number. Implements the formula:
```
  
   log(a + bi) = ln(|a + bi|) + arg(a + bi)i
  
 
```
    where ln on the right hand side is Math.log(double), |a + bi| is the modulus, abs(), and arg(a + bi) = Math.atan2(double, double)(b, a).
    Returns:
    
    the value ln this, the natural logarithm of this.
  - log10
```
public Complex log10()
```
    Compute the base 10 or common logarithm of this complex number.
    
    Returns:
    
    the base 10 logarithm of this.
  - pow
```
public Complex pow(Complex x)
```
    Returns of value of this complex number raised to the power of x. Implements the formula:
```
  
   y^x = exp(x·log(y))
  
 
```
    where exp and log are exp() and log(), respectively.
    Parameters:
    
    x - exponent to which this Complex is to be raised.
    
    Returns:
    
    this^x.
  - pow
```
public Complex pow(double x)
```
    Returns of value of this complex number raised to the power of x.
    
    Parameters:
    
    x - exponent to which this Complex is to be raised.
    
    Returns:
    
    this^x.
    
    See Also:
    
    pow(Complex)
  - sin
```
public Complex sin()
```
    Compute the sine of this complex number. Implements the formula:
```
  
   sin(a + bi) = sin(a)cosh(b) - cos(a)sinh(b)i
  
 
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), Math.cosh(double) and Math.sinh(double).
    Returns:
    
    the sine of this complex number.
  - sinh
```
public Complex sinh()
```
    Compute the hyperbolic sine of this complex number. Implements the formula:
```
  
   sinh(a + bi) = sinh(a)cos(b)) + cosh(a)sin(b)i
  
 
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), Math.cosh(double) and Math.sinh(double).
    Returns:
    
    the hyperbolic sine of this.
  - sqrt
```
public Complex sqrt()
```
    Compute the square root of this complex number. Implements the following algorithm to compute sqrt(a + bi):
    1. Let t = sqrt((|a| + |a + bi|) / 2)
    2. if a ≥ 0 return t + (b/2t)i else return |b|/2t + sign(b)t i
    where
    - |a| = Math.abs(int)(a)
    - |a + bi| = abs()(a + bi)
    - sign(b) = copySign(1d, b)
    Returns:
    
    the square root of this.
  - tan
```
public Complex tan()
```
    Compute the tangent of this complex number. Implements the formula:
```
  
   tan(a + bi) = sin(2a)/(cos(2a)+cosh(2b)) + [sinh(2b)/(cos(2a)+cosh(2b))]i
  
 
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), Math.cosh(double) and Math.sinh(double).
    Returns:
    
    the tangent of this.
  - tanh
```
public Complex tanh()
```
    Compute the hyperbolic tangent of this complex number. Implements the formula:
```
  
   tan(a + bi) = sinh(2a)/(cosh(2a)+cos(2b)) + [sin(2b)/(cosh(2a)+cos(2b))]i
  
 
```
    where the (real) functions on the right-hand side are Math.sin(double), Math.cos(double), Math.cosh(double) and Math.sinh(double).
    Returns:
    
    the hyperbolic tangent of this.
  - getArgument
```
public double getArgument()
```
    Compute the argument of this complex number. The argument is the angle phi between the positive real axis and the point representing this number in the complex plane. The value returned is between -PI (not inclusive) and PI (inclusive), with negative values returned for numbers with negative imaginary parts.
    If either real or imaginary part (or both) is NaN, NaN is returned. Infinite parts are handled as Math.atan2 handles them, essentially treating finite parts as zero in the presence of an infinite coordinate and returning a multiple of pi/4 depending on the signs of the infinite parts. See the javadoc for Math.atan2 for full details.
    
    Returns:
    
    the argument of this.
  - arg
```
public double arg()
```
    Compute the argument of this complex number. C++11 syntax
    
    Returns:
    
    the argument of this.
  - nthRoot
```
public List<Complex> nthRoot(int n)
```
    Computes the n-th roots of this complex number. The nth roots are defined by the formula:
```
  
   z_k = abs^1/n (cos(phi + 2πk/n) + i (sin(phi + 2πk/n))
  
 
```
    for k=0, 1, ..., n-1, where abs and phi are respectively the modulus and argument of this complex number.
    If one or both parts of this complex number is NaN, a list with just one element, NaN + NaN i is returned. if neither part is NaN, but at least one part is infinite, the result is a one-element list containing INF.
    Parameters:
    
    n - Degree of root.
    
    Returns:
    
    a List of all n-th roots of this.
  - toString
```
public String toString()
```
    Overrides:
    
    toString in class Object

Class Complex

Field Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

I

INF

ONE

ZERO

Method Detail

ofCartesian

ofReal

ofPolar

ofCis

parse

isNaN

isInfinite

proj

abs

add

add

conjugate

conj

divide

divide

reciprocal

equals

equals

equals

equals

equalsWithRelativeTolerance

hashCode

getImaginary

imag

getReal

real

multiply

multiply

multiply

negate

subtract

subtract

acos

asin

atan

asinh

atanh

acosh

square

cos

cosh

exp

log

log10

pow

pow

sin

sinh

sqrt

tan

tanh

getArgument

arg

nthRoot

toString