org.apache.commons.math3.complex

## Class Complex

- java.lang.Object
- - org.apache.commons.math3.complex.Complex

All Implemented Interfaces:
  Serializable, FieldElement<Complex>

    public class Complex
    extends Object
    implements FieldElement<Complex>, 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 is in contradiction with 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 `Precision` conforms with IEEE-754 while this class conforms with the standard behavior for Java object types.
Implements Serializable since 2.0

Version:
  $Id: Complex.java 1416643 2012-12-03 19:37:14Z tn $
See Also:
  Serialized Form

### Field Summary

Fields
`static Complex` `I` The square root of -1.
`private double` `imaginary` The imaginary part.
`static Complex` `INF` A complex number representing "+INF + INFi"
`private boolean` `isInfinite` Record whether this complex number is infinite.
`private boolean` `isNaN` Record whether this complex number is equal to NaN.
`static Complex` `NaN` A complex number representing "NaN + NaNi"
`static Complex` `ONE` A complex number representing "1.0 + 0.0i"
`private double` `real` The real part.
`private static long` `serialVersionUID` Serializable version identifier
`static Complex` `ZERO` A complex number representing "0.0 + 0.0i"

### Constructor Summary

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

### Method Summary

Methods
`double` `abs()` Return the absolute value of this complex number.
`Complex` `acos()` Compute the inverse 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.
`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.
`protected Complex` `createComplex(double realPart, double imaginaryPart)` Create a complex number given the real and imaginary parts.
`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.
`boolean` `equals(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()` Checks whether 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()` Checks whether either or both parts of this complex number is `NaN`.
`Complex` `log()` Compute the natural 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.
`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`.
`protected Object` `readResolve()` Resolve the transient fields in a deserialized Complex Object.
`Complex` `reciprocal()` Returns the multiplicative inverse of `this` element.
`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 - this2` for 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()`
`static Complex` `valueOf(double realPart)` Create a complex number given only the real part.
`static Complex` `valueOf(double realPart, double imaginaryPart)` Create a complex number given the real and imaginary parts.

### 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 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"

#### serialVersionUID

    private static final long serialVersionUID

Serializable version identifier

See Also:
  Constant Field Values

#### imaginary

    private final double imaginary

The imaginary part.

#### real

    private final double real

The real part.

#### isNaN

    private final transient boolean isNaN

Record whether this complex number is equal to NaN.

#### isInfinite

    private final transient boolean isInfinite

Record whether this complex number is infinite.

### Constructor Detail

#### Complex

    public Complex(double real)

Create a complex number given only the real part.

Parameters:
  `real` - Real part.

#### Complex

    public Complex(double real,
           double imaginary)

Create a complex number given the real and imaginary parts.

Parameters:
  `real` - Real part.
  `imaginary` - 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 is infinite.

Returns:
  the absolute value.

#### add

    public Complex add(Complex addend)
                throws NullArgumentException

Returns a `Complex` whose value is `(this + addend)`. Uses the definitional formula

       (a + bi) + (c + di) = (a+c) + (b+d)i

If either `this` or `addend` has a `NaN` value in either part, `NaN` is returned; otherwise `Infinite` 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:
  `addend` - Value to be added to this `Complex`.
Returns:
  `this + addend`.
Throws:
  `NullArgumentException` - if `addend` is `null`.

#### 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()

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 divisor)
                   throws NullArgumentException

Returns a `Complex` whose value is `(this / divisor)`. 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 according to the following rules, applied in the order presented:

- If either `this` or `divisor` has a `NaN` value in either part, `NaN` is returned.
- If `divisor` equals `ZERO`, `NaN` is returned.
- If `this` and `divisor` are both infinite, `NaN` is returned.
- If `this` is finite (i.e., has no `Infinite` or `NaN` parts) and `divisor` is infinite (one or both parts infinite), `ZERO` is returned.
- If `this` is infinite and `divisor` 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:
  `divisor` - Value by which this `Complex` is to be divided.
Returns:
  `this / divisor`.
Throws:
  `NullArgumentException` - if `divisor` is `null`.

#### 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` element.

Specified by:
  `reciprocal` in interface `FieldElement<Complex>`
Returns:
  the inverse of `this`.

#### equals

    public boolean equals(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 `NaN`.

Overrides:
  `equals` in class `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. 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.

#### getReal

    public double getReal()

Access the real part.

Returns:
  the real part.

#### isNaN

    public boolean isNaN()

Checks whether either or both parts of this complex number is `NaN`.

Returns:
  true if either or both parts of this complex number is `NaN`; false otherwise.

#### isInfinite

    public boolean isInfinite()

Checks whether 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 factor)
                     throws NullArgumentException

Returns a `Complex` whose value is `this * factor`. 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 `factor` has one or more `NaN` parts.
Returns `INF` if neither `this` nor `factor` has one or more `NaN` parts and if either `this` or `factor` 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:
  `factor` - value to be multiplied by this `Complex`.
Returns:
  `this * factor`.
Throws:
  `NullArgumentException` - if `factor` is `null`.

#### multiply

    public Complex multiply(int factor)

Returns a `Complex` whose value is `this * factor`, with `factor` interpreted as a integer number.

Specified by:
  `multiply` in interface `FieldElement<Complex>`
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 `NaN` if either real or imaginary part of this Complex number equals `Double.NaN`.

Specified by:
  `negate` in interface `FieldElement<Complex>`
Returns:
  `-this`.

#### subtract

    public Complex subtract(Complex subtrahend)
                     throws NullArgumentException

Returns a `Complex` whose value is `(this - subtrahend)`. Uses the definitional formula

       (a + bi) - (c + di) = (a-c) + (b-d)i

If either `this` or `subtrahend` has a `NaN]` value in either part, `NaN` is returned; otherwise infinite 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:
  `subtrahend` - value to be subtracted from this `Complex`.
Returns:
  `this - subtrahend`.
Throws:
  `NullArgumentException` - if `subtrahend` is `null`.

#### 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 - 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)`, `FastMath.cosh(double)` and `FastMath.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)`, `FastMath.cosh(double)` and `FastMath.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:
  `ethis`.
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:
  the value `ln this`, the natural logarithm of `this`.
Since:
  1.2

#### pow

    public Complex pow(Complex x)
                throws NullArgumentException

Returns of value of this complex number raised to the power of `x`. Implements the formula:

       yx = 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` - exponent to which this `Complex` is to be raised.
Returns:
  `this`x``.
Throws:
  `NullArgumentException` - if x is `null`.
Since:
  1.2

#### 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:
  `thisx`.
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)`, `FastMath.cosh(double)` and `FastMath.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)`, `FastMath.cosh(double)` and `FastMath.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`.
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 &#8805; 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 `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`.
Since:
  1.2

#### sqrt1z

    public Complex sqrt1z()

Compute the square root of `1 - this2` for this complex number. Computes the result directly as `sqrt(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 `FastMath.sin(double)`, `FastMath.cos(double)`, `FastMath.cosh(double)` and `FastMath.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(a ± INFINITY i) = 0 ± i
       tan(±INFINITY + bi) = NaN + NaN i
       tan(±INFINITY ± INFINITY i) = NaN + NaN i
       tan(±π/2 + 0 i) = ±INFINITY + NaN i

Returns:
  the tangent of `this`.
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 `FastMath.sin(double)`, `FastMath.cos(double)`, `FastMath.cosh(double)` and `FastMath.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(a ± INFINITY i) = NaN + NaN i
       tanh(±INFINITY + bi) = ±1 + 0 i
       tanh(±INFINITY ± INFINITY i) = NaN + NaN i
       tanh(0 + (π/2)i) = NaN + INFINITY i

Returns:
  the hyperbolic tangent of `this`.
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 `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`.

#### nthRoot

    public List<Complex> nthRoot(int n)
                          throws NotPositiveException

Computes the n-th roots of this complex number. The nth roots are defined by the formula:

       zk = abs1/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:
  a List<Complex> of all `n`-th roots of `this`.
Throws:
  `NotPositiveException` - if `n <= 0`.
Since:
  2.0

#### createComplex

    protected Complex createComplex(double realPart,
                        double imaginaryPart)

Create a complex number given the real and imaginary parts.

Parameters:
  `realPart` - Real part.
  `imaginaryPart` - Imaginary part.
Returns:
  a new complex number instance.
Since:
  1.2
See Also:
  `valueOf(double, double)`

#### valueOf

    public static Complex valueOf(double realPart,
                  double imaginaryPart)

Create a complex number given the real and imaginary parts.

Parameters:
  `realPart` - Real part.
  `imaginaryPart` - Imaginary part.
Returns:
  a Complex instance.

#### valueOf

    public static Complex valueOf(double realPart)

Create a complex number given only the real part.

Parameters:
  `realPart` - Real part.
Returns:
  a Complex instance.

#### readResolve

    protected final Object readResolve()

Resolve the transient fields in a deserialized Complex Object. Subclasses will need to override `createComplex(double, double)` to deserialize properly.

Returns:
  A Complex instance with all fields resolved.
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

#### toString

    public String toString()

Overrides:
  `toString` in class `Object`