Complex (Apache Commons Math 3.6.1 API)

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 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 `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.
    `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)` 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
    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 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()` 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.
    `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.
    `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.
    `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"

  - ### 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()

      Returns 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 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`.
      Since:
        3.3
      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.
      Since:
        3.3

    - #### 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.
      Since:
        3.3
      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.
      Since:
        3.3
      See Also:
        `Precision.equalsWithRelativeTolerance(double,double,double)`

    - #### 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 is `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 - z<sup>2</sup>))))`

      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 - z<sup>2</sup>) + 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 `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:

             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 `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:

             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 `FastMath.exp(double)`, `FastMath.cos(double)`, and `FastMath.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 `FastMath.log(double)`, `|a + bi|` is the modulus, `abs()`, and `arg(a + bi) =` `FastMath.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:
        `thisx`.
      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 `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:

             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 `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:

             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| =` `FastMath.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 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`

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