Complex (The Adobe AEM Quickstart and Web Application.)

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

All Implemented Interfaces:

java.io.Serializable, FieldElement<Complex>
```
public class Complex
extends java.lang.Object
implements FieldElement<Complex>, java.io.Serializable
```
Representation of a Complex number - 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 arithmetic, applying definitional formulas and returning NaN or infinite values in real or imaginary parts as these arise in computation. See individual method javadocs for details.

equals(java.lang.Object) identifies all values with NaN in either real or imaginary part - e.g.,
```
 1 + NaNi  == NaN + i == NaN + NaNi.
```
implements Serializable since 2.0
See Also:

Serialized Form

Field Summary

Fields
Modifier and Type	Field and Description
`static Complex`	`I` The square root of -1.
`static Complex`	`INF` A complex number representing "+INF + INFi"
`static Complex`	`NaN` A complex number representing "NaN + NaNi"
`static Complex`	`ONE` A complex number representing "1.0 + 0.0i"
`static Complex`	`ZERO` A complex number representing "0.0 + 0.0i"

Constructor Summary

Constructors
Constructor and Description

Complex(double real, double imaginary)
Create a complex number given the real and imaginary parts.

Constructors
Constructor and Description
`Complex(double real, double imaginary)` Create a complex number given the real and imaginary parts.

Method Summary

All 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`	`add(Complex rhs)` Return the sum of this complex number and the given complex number.
`Complex`	`asin()` Compute the inverse sine of this complex number.
`Complex`	`atan()` Compute the inverse tangent of this complex number.
`Complex`	`conjugate()` Return 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 rhs)` Return the quotient of this complex number and the given complex number.
`boolean`	`equals(java.lang.Object other)` Test for the equality of two Complex objects.
`Complex`	`exp()` Compute the exponential function of this complex number.
`double`	`getArgument()` Compute the argument of this complex number.
`ComplexField`	`getField()` Get the `Field` to which the instance belongs.
`double`	`getImaginary()` Access the imaginary part.
`double`	`getReal()` Access the real part.
`int`	`hashCode()` Get a hashCode for the complex number.
`boolean`	`isInfinite()` Returns true if either the real or imaginary part of this complex number takes an infinite value (either `Double.POSITIVE_INFINITY` or `Double.NEGATIVE_INFINITY`) and neither part is `NaN`.
`boolean`	`isNaN()` Returns true if either or both parts of this complex number is NaN; false otherwise
`Complex`	`log()` Compute the natural logarithm of this complex number.
`Complex`	`multiply(Complex rhs)` Return the product of this complex number and the given complex number.
`Complex`	`multiply(double rhs)` Return the product of this complex number and the given scalar number.
`Complex`	`negate()` Return the additive inverse of this complex number.
`java.util.List<Complex>`	`nthRoot(int n)` Computes the n-th roots of this complex number.
`Complex`	`pow(Complex x)` Returns of value of this complex number raised to the power of `x`.
`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`	`sqrt1z()` Compute the square root of 1 - `this`² for this complex number.
`Complex`	`subtract(Complex rhs)` Return the difference between this complex number and the given complex number.
`Complex`	`tan()` Compute the tangent of this complex number.
`Complex`	`tanh()` Compute the hyperbolic tangent of this complex number.

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

- Field Detail
  - I
```
public static final Complex I
```
    The square root of -1. A number representing "0.0 + 1.0i"
  - NaN
```
public static final Complex NaN
```
    A complex number representing "NaN + NaNi"
  - INF
```
public static final Complex INF
```
    A complex number representing "+INF + INFi"
  - ONE
```
public static final Complex ONE
```
    A complex number representing "1.0 + 0.0i"
  - ZERO
```
public static final Complex ZERO
```
    A complex number representing "0.0 + 0.0i"
- Constructor Detail
  - Complex
```
public Complex(double real,
               double imaginary)
```
    Create a complex number given the real and imaginary parts.
    
    Parameters:
    
    real - the real part
    
    imaginary - the imaginary part
- Method Detail
  - abs
```
public double abs()
```
    Return the absolute value of this complex number.
    Returns NaN if either real or imaginary part is NaN and Double.POSITIVE_INFINITY if neither part is NaN, but at least one part takes an infinite value.
    
    Returns:
    
    the absolute value
  - add
```
public Complex add(Complex rhs)
```
    Return the sum of this complex number and the given complex number.
    Uses the definitional formula
```
 (a + bi) + (c + di) = (a+c) + (b+d)i
 
```
    If either this or rhs has a NaN value in either part, NaN is returned; otherwise Inifinite and NaN values are returned in the parts of the result according to the rules for Double arithmetic.
    Specified by:
    
    add in interface FieldElement<Complex>
    
    Parameters:
    
    rhs - the other complex number
    
    Returns:
    
    the complex number sum
    
    Throws:
    
    java.lang.NullPointerException - if rhs is null
  - conjugate
```
public Complex conjugate()
```
    Return the conjugate of this complex number. The conjugate of "A + Bi" is "A - Bi".
    NaN is returned if either the real or imaginary part of this Complex number equals Double.NaN.
    
    If the imaginary part is infinite, and the real part is not NaN, the returned value has infinite imaginary part of the opposite sign - e.g. the conjugate of 1 + POSITIVE_INFINITY i is 1 - NEGATIVE_INFINITY i
    
    Returns:
    
    the conjugate of this Complex object
  - divide
```
public Complex divide(Complex rhs)
```
    Return the quotient of this complex number and the given complex number.
    Implements the definitional formula
```
    a + bi          ac + bd + (bc - ad)i
    ----------- = -------------------------
    c + di         c² + d²
 
```
    but uses prescaling of operands to limit the effects of overflows and underflows in the computation.
    
    Infinite and NaN values are handled / returned according to the following rules, applied in the order presented:
    - If either this or rhs has a NaN value in either part, NaN is returned.
    - If rhs equals ZERO, NaN is returned.
    - If this and rhs are both infinite, NaN is returned.
    - If this is finite (i.e., has no infinite or NaN parts) and rhs is infinite (one or both parts infinite), ZERO is returned.
    - If this is infinite and rhs is finite, NaN values are returned in the parts of the result if the Double rules applied to the definitional formula force NaN results.
    Specified by:
    
    divide in interface FieldElement<Complex>
    
    Parameters:
    
    rhs - the other complex number
    
    Returns:
    
    the complex number quotient
    
    Throws:
    
    java.lang.NullPointerException - if rhs is null
  - equals
```
public boolean equals(java.lang.Object other)
```
    Test for the equality of two Complex objects.
    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.
    
    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 Complex.NaN.
    
    Overrides:
    
    equals in class java.lang.Object
    
    Parameters:
    
    other - Object to test for equality to this
    
    Returns:
    
    true if two Complex objects are equal, false if object is null, not an instance of Complex, or not equal to this Complex instance
  - hashCode
```
public int hashCode()
```
    Get a hashCode for the complex number.
    All NaN values have the same hash code.
    
    Overrides:
    
    hashCode in class java.lang.Object
    
    Returns:
    
    a hash code value for this object
  - getImaginary
```
public double getImaginary()
```
    Access the imaginary part.
    
    Returns:
    
    the imaginary part
  - getReal
```
public double getReal()
```
    Access the real part.
    
    Returns:
    
    the real part
  - isNaN
```
public boolean isNaN()
```
    Returns true if either or both parts of this complex number is NaN; false otherwise
    
    Returns:
    
    true if either or both parts of this complex number is NaN; false otherwise
  - isInfinite
```
public boolean isInfinite()
```
    Returns true if either the real or imaginary part of this complex number takes an infinite value (either Double.POSITIVE_INFINITY or Double.NEGATIVE_INFINITY) and neither part is NaN.
    
    Returns:
    
    true if one or both parts of this complex number are infinite and neither part is NaN
  - multiply
```
public Complex multiply(Complex rhs)
```
    Return the product of this complex number and the given complex number.
    Implements preliminary checks for NaN and infinity followed by the definitional formula:
```
 (a + bi)(c + di) = (ac - bd) + (ad + bc)i
 
```
    Returns NaN if either this or rhs has one or more NaN parts.
    Returns INF if neither this nor rhs has one or more NaN parts and if either this or rhs has one or more infinite parts (same result is returned regardless of the sign of the components).
    
    Returns finite values in components of the result per the definitional formula in all remaining cases.
    Specified by:
    
    multiply in interface FieldElement<Complex>
    
    Parameters:
    
    rhs - the other complex number
    
    Returns:
    
    the complex number product
    
    Throws:
    
    java.lang.NullPointerException - if rhs is null
  - multiply
```
public Complex multiply(double rhs)
```
    Return the product of this complex number and the given scalar number.
    Implements preliminary checks for NaN and infinity followed by the definitional formula:
```
 c(a + bi) = (ca) + (cb)i
 
```
    Returns NaN if either this or rhs has one or more NaN parts.
    Returns INF if neither this nor rhs has one or more NaN parts and if either this or rhs has one or more infinite parts (same result is returned regardless of the sign of the components).
    
    Returns finite values in components of the result per the definitional formula in all remaining cases.
    Parameters:
    
    rhs - the scalar number
    
    Returns:
    
    the complex number product
  - negate
```
public Complex negate()
```
    Return the additive inverse of this complex number.
    Returns Complex.NaN if either real or imaginary part of this Complex number equals Double.NaN.
    
    Returns:
    
    the negation of this complex number
  - subtract
```
public Complex subtract(Complex rhs)
```
    Return the difference between this complex number and the given complex number.
    Uses the definitional formula
```
 (a + bi) - (c + di) = (a-c) + (b-d)i
 
```
    If either this or rhs has a NaN value in either part, NaN is returned; otherwise inifinite and NaN values are returned in the parts of the result according to the rules for Double arithmetic.
    Specified by:
    
    subtract in interface FieldElement<Complex>
    
    Parameters:
    
    rhs - the other complex number
    
    Returns:
    
    the complex number difference
    
    Throws:
    
    java.lang.NullPointerException - if rhs is null
  - acos
```
public Complex acos()
```
    Compute the inverse cosine of this complex number.
    Implements the formula:
```
  acos(z) = -i (log(z + i (sqrt(1 - z²))))
```
    Returns NaN if either real or imaginary part of the input argument is NaN or infinite.
    Returns:
    
    the inverse cosine of this complex number
    
    Since:
    
    1.2
  - asin
```
public Complex asin()
```
    Compute the inverse sine of this complex number.
    Implements the formula:
```
  asin(z) = -i (log(sqrt(1 - z²) + iz)) 
```
    Returns NaN if either real or imaginary part of the input argument is NaN or infinite.
    Returns:
    
    the inverse sine of this complex number.
    
    Since:
    
    1.2
  - 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 NaN if either real or imaginary part of the input argument is NaN or infinite.
    Returns:
    
    the inverse tangent of this complex number
    
    Since:
    
    1.2
  - 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), MathUtils.cosh(double) and MathUtils.sinh(double).
    
    Returns NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 cos(1 ± INFINITY i) = 1 ∓ INFINITY i
 cos(±INFINITY + i) = NaN + NaN i
 cos(±INFINITY ± INFINITY i) = NaN + NaN i
```
    Returns:
    
    the cosine of this complex number
    
    Since:
    
    1.2
  - 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), MathUtils.cosh(double) and MathUtils.sinh(double).
    
    Returns NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 cosh(1 ± INFINITY i) = NaN + NaN i
 cosh(±INFINITY + i) = INFINITY ± INFINITY i
 cosh(±INFINITY ± INFINITY i) = NaN + NaN i
```
    Returns:
    
    the hyperbolic cosine of this complex number.
    
    Since:
    
    1.2
  - 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 NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 exp(1 ± INFINITY i) = NaN + NaN i
 exp(INFINITY + i) = INFINITY + INFINITY i
 exp(-INFINITY + i) = 0 + 0i
 exp(±INFINITY ± INFINITY i) = NaN + NaN i
```
    Returns:
    
    e^this
    
    Since:
    
    1.2
  - 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 NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite (or critical) values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 log(1 ± INFINITY i) = INFINITY ± (π/2)i
 log(INFINITY + i) = INFINITY + 0i
 log(-INFINITY + i) = INFINITY + πi
 log(INFINITY ± INFINITY i) = INFINITY ± (π/4)i
 log(-INFINITY ± INFINITY i) = INFINITY ± (3π/4)i
 log(0 + 0i) = -INFINITY + 0i
 
```
    Returns:
    
    ln of this complex number.
    
    Since:
    
    1.2
  - 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.
    
    Returns NaN if either real or imaginary part of the input argument is NaN or infinite, or if y equals ZERO.
    Parameters:
    
    x - the exponent.
    
    Returns:
    
    this^x
    
    Throws:
    
    java.lang.NullPointerException - if x is null
    
    Since:
    
    1.2
  - 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), MathUtils.cosh(double) and MathUtils.sinh(double).
    
    Returns NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 sin(1 ± INFINITY i) = 1 ± INFINITY i
 sin(±INFINITY + i) = NaN + NaN i
 sin(±INFINITY ± INFINITY i) = NaN + NaN i
```
    Returns:
    
    the sine of this complex number.
    
    Since:
    
    1.2
  - 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), MathUtils.cosh(double) and MathUtils.sinh(double).
    
    Returns NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 sinh(1 ± INFINITY i) = NaN + NaN i
 sinh(±INFINITY + i) = ± INFINITY + INFINITY i
 sinh(±INFINITY ± INFINITY i) = NaN + NaN i
```
    Returns:
    
    the hyperbolic sine of this complex number
    
    Since:
    
    1.2
  - 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) = MathUtils.indicator(byte)(b)
    Returns NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 sqrt(1 ± INFINITY i) = INFINITY + NaN i
 sqrt(INFINITY + i) = INFINITY + 0i
 sqrt(-INFINITY + i) = 0 + INFINITY i
 sqrt(INFINITY ± INFINITY i) = INFINITY + NaN i
 sqrt(-INFINITY ± INFINITY i) = NaN ± INFINITY i
 
```
    Returns:
    
    the square root of this complex number
    
    Since:
    
    1.2
  - sqrt1z
```
public Complex sqrt1z()
```
    Compute the square root of 1 - this² for this complex number.
    Computes the result directly as sqrt(Complex.ONE.subtract(z.multiply(z))).
    
    Returns NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
    
    Returns:
    
    the square root of 1 - this²
    
    Since:
    
    1.2
  - 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), MathUtils.cosh(double) and MathUtils.sinh(double).
    
    Returns NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite (or critical) values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 tan(1 ± INFINITY i) = 0 + NaN i
 tan(±INFINITY + i) = NaN + NaN i
 tan(±INFINITY ± INFINITY i) = NaN + NaN i
 tan(±π/2 + 0 i) = ±INFINITY + NaN i
```
    Returns:
    
    the tangent of this complex number
    
    Since:
    
    1.2
  - 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), MathUtils.cosh(double) and MathUtils.sinh(double).
    
    Returns NaN if either real or imaginary part of the input argument is NaN.
    
    Infinite values in real or imaginary parts of the input may result in infinite or NaN values returned in parts of the result.
```
 Examples:
 
 tanh(1 ± INFINITY i) = NaN + NaN i
 tanh(±INFINITY + i) = NaN + 0 i
 tanh(±INFINITY ± INFINITY i) = NaN + NaN i
 tanh(0 + (π/2)i) = NaN + INFINITY i
```
    Returns:
    
    the hyperbolic tangent of this complex number
    
    Since:
    
    1.2
  - 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 java.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 java.Math.atan2 for full details.
    
    Returns:
    
    the argument of this complex number
  - nthRoot
```
public java.util.List<Complex> nthRoot(int n)
                                throws java.lang.IllegalArgumentException
```
    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 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:
    
    List all nth roots of this complex number
    
    Throws:
    
    java.lang.IllegalArgumentException - if parameter n is less than or equal to 0
    
    Since:
    
    2.0
  - getField
```
public ComplexField getField()
```
    Get the Field to which the instance belongs.
    
    Specified by:
    
    getField in interface FieldElement<Complex>
    
    Returns:
    
    Field to which the instance belongs

Class Complex

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

I

NaN

INF

ONE

ZERO

Constructor Detail

Complex

Method Detail

abs

add

conjugate

divide

equals

hashCode

getImaginary

getReal

isNaN

isInfinite

multiply

multiply

negate

subtract

acos

asin

atan

cos

cosh

exp

log

pow

sin

sinh

sqrt

sqrt1z

tan

tanh

getArgument

nthRoot

getField