/*
    * Licensed to the Apache Software Foundation (ASF) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * The ASF licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *      https://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  
  /*
   * This is not the original file distributed by the Apache Software Foundation
   * It has been modified by the Hipparchus project
   */
  
  package org.hipparchus.complex;
  
  import java.io.Serializable;
  import java.util.ArrayList;
  import java.util.List;
  
  import org.hipparchus.CalculusFieldElement;
  import org.hipparchus.exception.LocalizedCoreFormats;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.exception.NullArgumentException;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.FieldSinCos;
  import org.hipparchus.util.FieldSinhCosh;
  import org.hipparchus.util.MathArrays;
  import org.hipparchus.util.MathUtils;
  import org.hipparchus.util.Precision;
  import org.hipparchus.util.SinCos;
  import org.hipparchus.util.SinhCosh;
  
  /**
   * Representation of a Complex number, i.e. a number which has both a
   * real and imaginary part.
   * <p>
   * Implementations of arithmetic operations handle {@code NaN} and
   * infinite values according to the rules for {@link java.lang.Double}, i.e.
   * {@link #equals} is an equivalence relation for all instances that have
   * a {@code NaN} in either real or imaginary part, e.g. the following are
   * considered equal:
   * <ul>
   *  <li>{@code 1 + NaNi}</li>
   *  <li>{@code NaN + i}</li>
   *  <li>{@code NaN + NaNi}</li>
   * </ul>
   * <p>
   * Note that this contradicts the IEEE-754 standard for floating
   * point numbers (according to which the test {@code x == x} must fail if
   * {@code x} is {@code NaN}). The method
   * {@link org.hipparchus.util.Precision#equals(double,double,int)
   * equals for primitive double} in {@link org.hipparchus.util.Precision}
   * conforms with IEEE-754 while this class conforms with the standard behavior
   * for Java object types.
   */
  public class Complex implements CalculusFieldElement<Complex>, Comparable<Complex>, Serializable  {
      /** The square root of -1. A number representing "0.0 + 1.0i". */
      public static final Complex I = new Complex(0.0, 1.0);
      /** The square root of -1. A number representing "0.0 - 1.0i".
       * @since 1.7
       */
      public static final Complex MINUS_I = new Complex(0.0, -1.0);
      // CHECKSTYLE: stop ConstantName
      /** A complex number representing "NaN + NaNi". */
      public static final Complex NaN = new Complex(Double.NaN, Double.NaN);
      // CHECKSTYLE: resume ConstantName
      /** A complex number representing "+INF + INFi" */
      public static final Complex INF = new Complex(Double.POSITIVE_INFINITY, Double.POSITIVE_INFINITY);
      /** A complex number representing "1.0 + 0.0i". */
      public static final Complex ONE = new Complex(1.0, 0.0);
      /** A complex number representing "-1.0 + 0.0i".
       * @since 1.7
       */
      public static final Complex MINUS_ONE = new Complex(-1.0, 0.0);
      /** A complex number representing "0.0 + 0.0i". */
      public static final Complex ZERO = new Complex(0.0, 0.0);
      /** A complex number representing "π + 0.0i". */
      public static final Complex PI   = new Complex(FastMath.PI, 0.0);
  
      /** A real number representing log(10). */
      private static final double LOG10 = 2.302585092994045684;
  
      /** Serializable version identifier */
      private static final long serialVersionUID = 20160305L;
  
      /** The imaginary part. */
      private final double imaginary;
      /** The real part. */
      private final double real;
      /** Record whether this complex number is equal to NaN. */
     private final transient boolean isNaN;
     /** Record whether this complex number is infinite. */
     private final transient boolean isInfinite;
 
     /**
      * Create a complex number given only the real part.
      *
      * @param real Real part.
      */
     public Complex(double real) {
         this(real, 0.0);
     }
 
     /**
      * Create a complex number given the real and imaginary parts.
      *
      * @param real Real part.
      * @param imaginary Imaginary part.
      */
     public Complex(double real, double imaginary) {
         this.real = real;
         this.imaginary = imaginary;
 
         isNaN = Double.isNaN(real) || Double.isNaN(imaginary);
         isInfinite = !isNaN &&
             (Double.isInfinite(real) || Double.isInfinite(imaginary));
     }
 
     /**
      * Return the absolute value of this complex number.
      * Returns {@code NaN} if either real or imaginary part is {@code NaN}
      * and {@code Double.POSITIVE_INFINITY} if neither part is {@code NaN},
      * but at least one part is infinite.
      *
      * @return the norm.
      * @since 2.0
      */
     @Override
     public Complex abs() {
         // we check NaN here because FastMath.hypot checks it after infinity
         return isNaN ? NaN : createComplex(FastMath.hypot(real, imaginary), 0.0);
     }
 
     /** {@inheritDoc} */
     @Override
     public double norm() {
         // we check NaN here because FastMath.hypot checks it after infinity
         return isNaN ? Double.NaN : FastMath.hypot(real, imaginary);
     }
 
     /**
      * Returns a {@code Complex} whose value is
      * {@code (this + addend)}.
      * Uses the definitional formula
      * <p>
      *   {@code (a + bi) + (c + di) = (a+c) + (b+d)i}
      * </p>
      * If either {@code this} or {@code addend} has a {@code NaN} value in
      * either part, {@link #NaN} is returned; otherwise {@code Infinite}
      * and {@code NaN} values are returned in the parts of the result
      * according to the rules for {@link java.lang.Double} arithmetic.
      *
      * @param  addend Value to be added to this {@code Complex}.
      * @return {@code this + addend}.
      * @throws NullArgumentException if {@code addend} is {@code null}.
      */
     @Override
     public Complex add(Complex addend) throws NullArgumentException {
         MathUtils.checkNotNull(addend);
         if (isNaN || addend.isNaN) {
             return NaN;
         }
 
         return createComplex(real + addend.getRealPart(),
                              imaginary + addend.getImaginaryPart());
     }
 
     /**
      * Returns a {@code Complex} whose value is {@code (this + addend)},
      * with {@code addend} interpreted as a real number.
      *
      * @param addend Value to be added to this {@code Complex}.
      * @return {@code this + addend}.
      * @see #add(Complex)
      */
     @Override
     public Complex add(double addend) {
         if (isNaN || Double.isNaN(addend)) {
             return NaN;
         }
 
         return createComplex(real + addend, imaginary);
     }
 
      /**
      * Returns the conjugate of this complex number.
      * The conjugate of {@code a + bi} is {@code a - bi}.
      * <p>
      * {@link #NaN} is returned if either the real or imaginary
      * part of this Complex number equals {@code Double.NaN}.
      * </p><p>
      * If the imaginary part is infinite, and the real part is not
      * {@code NaN}, the returned value has infinite imaginary part
      * of the opposite sign, e.g. the conjugate of
      * {@code 1 + POSITIVE_INFINITY i} is {@code 1 - NEGATIVE_INFINITY i}.
      * </p>
      * @return the conjugate of this Complex object.
      */
     public Complex conjugate() {
         if (isNaN) {
             return NaN;
         }
 
         return createComplex(real, -imaginary);
     }
 
     /**
      * Returns a {@code Complex} whose value is
      * {@code (this / divisor)}.
      * Implements the definitional formula
      * <pre>
      *  <code>
      *    a + bi          ac + bd + (bc - ad)i
      *    ----------- = -------------------------
      *    c + di         c<sup>2</sup> + d<sup>2</sup>
      *  </code>
      * </pre>
      * but uses
      * <a href="http://doi.acm.org/10.1145/1039813.1039814">
      * prescaling of operands</a> to limit the effects of overflows and
      * underflows in the computation.
      * <p>
      * {@code Infinite} and {@code NaN} values are handled according to the
      * following rules, applied in the order presented:
      * <ul>
      *  <li>If either {@code this} or {@code divisor} has a {@code NaN} value
      *   in either part, {@link #NaN} is returned.
      *  </li>
      *  <li>If {@code divisor} equals {@link #ZERO}, {@link #NaN} is returned.
      *  </li>
      *  <li>If {@code this} and {@code divisor} are both infinite,
      *   {@link #NaN} is returned.
      *  </li>
      *  <li>If {@code this} is finite (i.e., has no {@code Infinite} or
      *   {@code NaN} parts) and {@code divisor} is infinite (one or both parts
      *   infinite), {@link #ZERO} is returned.
      *  </li>
      *  <li>If {@code this} is infinite and {@code divisor} is finite,
      *   {@code NaN} values are returned in the parts of the result if the
      *   {@link java.lang.Double} rules applied to the definitional formula
      *   force {@code NaN} results.
      *  </li>
      * </ul>
      *
      * @param divisor Value by which this {@code Complex} is to be divided.
      * @return {@code this / divisor}.
      * @throws NullArgumentException if {@code divisor} is {@code null}.
      */
     @Override
     public Complex divide(Complex divisor)
         throws NullArgumentException {
         MathUtils.checkNotNull(divisor);
         if (isNaN || divisor.isNaN) {
             return NaN;
         }
 
         final double c = divisor.getRealPart();
         final double d = divisor.getImaginaryPart();
         if (c == 0.0 && d == 0.0) {
             return NaN;
         }
 
         if (divisor.isInfinite() && !isInfinite()) {
             return ZERO;
         }
 
         if (FastMath.abs(c) < FastMath.abs(d)) {
             double q = c / d;
             double denominator = c * q + d;
             return createComplex((real * q + imaginary) / denominator,
                                  (imaginary * q - real) / denominator);
         } else {
             double q = d / c;
             double denominator = d * q + c;
             return createComplex((imaginary * q + real) / denominator,
                                  (imaginary - real * q) / denominator);
         }
     }
 
     /**
      * Returns a {@code Complex} whose value is {@code (this / divisor)},
      * with {@code divisor} interpreted as a real number.
      *
      * @param  divisor Value by which this {@code Complex} is to be divided.
      * @return {@code this / divisor}.
      * @see #divide(Complex)
      */
     @Override
     public Complex divide(double divisor) {
         if (isNaN || Double.isNaN(divisor)) {
             return NaN;
         }
         if (divisor == 0d) {
             return NaN;
         }
         if (Double.isInfinite(divisor)) {
             return !isInfinite() ? ZERO : NaN;
         }
         return createComplex(real / divisor,
                              imaginary  / divisor);
     }
 
     /** {@inheritDoc} */
     @Override
     public Complex reciprocal() {
         if (isNaN) {
             return NaN;
         }
 
         if (real == 0.0 && imaginary == 0.0) {
             return INF;
         }
 
         if (isInfinite) {
             return ZERO;
         }
 
         if (FastMath.abs(real) < FastMath.abs(imaginary)) {
             double q = real / imaginary;
             double scale = 1. / (real * q + imaginary);
             return createComplex(scale * q, -scale);
         } else {
             double q = imaginary / real;
             double scale = 1. / (imaginary * q + real);
             return createComplex(scale, -scale * q);
         }
     }
 
     /**
      * Test for equality with another object.
      * If both the real and imaginary parts of two complex numbers
      * are exactly the same, and neither is {@code Double.NaN}, the two
      * Complex objects are considered to be equal.
      * The behavior is the same as for JDK's {@link Double#equals(Object)
      * Double}:
      * <ul>
      *  <li>All {@code NaN} values are considered to be equal,
      *   i.e, if either (or both) real and imaginary parts of the complex
      *   number are equal to {@code Double.NaN}, the complex number is equal
      *   to {@code NaN}.
      *  </li>
      *  <li>
      *   Instances constructed with different representations of zero (i.e.
      *   either "0" or "-0") are <em>not</em> considered to be equal.
      *  </li>
      * </ul>
      *
      * @param other Object to test for equality with this instance.
      * @return {@code true} if the objects are equal, {@code false} if object
      * is {@code null}, not an instance of {@code Complex}, or not equal to
      * this instance.
      */
     @Override
     public boolean equals(Object other) {
         if (this == other) {
             return true;
         }
         if (other instanceof Complex){
             Complex c = (Complex) other;
             if (c.isNaN) {
                 return isNaN;
             } else {
                 return MathUtils.equals(real, c.real) &&
                        MathUtils.equals(imaginary, c.imaginary);
             }
         }
         return false;
     }
 
     /**
      * Test for the floating-point equality between Complex objects.
      * It returns {@code true} if both arguments are equal or within the
      * range of allowed error (inclusive).
      *
      * @param x First value (cannot be {@code null}).
      * @param y Second value (cannot be {@code null}).
      * @param maxUlps {@code (maxUlps - 1)} is the number of floating point
      * values between the real (resp. imaginary) parts of {@code x} and
      * {@code y}.
      * @return {@code true} if there are fewer than {@code maxUlps} floating
      * point values between the real (resp. imaginary) parts of {@code x}
      * and {@code y}.
      *
      * @see Precision#equals(double,double,int)
      */
     public static boolean equals(Complex x, Complex y, int maxUlps) {
         return Precision.equals(x.real, y.real, maxUlps) &&
                Precision.equals(x.imaginary, y.imaginary, maxUlps);
     }
 
     /**
      * Returns {@code true} iff the values are equal as defined by
      * {@link #equals(Complex,Complex,int) equals(x, y, 1)}.
      *
      * @param x First value (cannot be {@code null}).
      * @param y Second value (cannot be {@code null}).
      * @return {@code true} if the values are equal.
      */
     public static boolean equals(Complex x, Complex y) {
         return equals(x, y, 1);
     }
 
     /**
      * Returns {@code 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 {@code false} if either of the arguments is NaN.
      *
      * @param x First value (cannot be {@code null}).
      * @param y Second value (cannot be {@code null}).
      * @param eps Amount of allowed absolute error.
      * @return {@code true} if the values are two adjacent floating point
      * numbers or they are within range of each other.
      *
      * @see Precision#equals(double,double,double)
      */
     public static boolean equals(Complex x, Complex y, double eps) {
         return Precision.equals(x.real, y.real, eps) &&
                Precision.equals(x.imaginary, y.imaginary, eps);
     }
 
     /**
      * Returns {@code 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 {@code false} if either of the arguments is NaN.
      *
      * @param x First value (cannot be {@code null}).
      * @param y Second value (cannot be {@code null}).
      * @param eps Amount of allowed relative error.
      * @return {@code true} if the values are two adjacent floating point
      * numbers or they are within range of each other.
      *
      * @see Precision#equalsWithRelativeTolerance(double,double,double)
      */
     public static boolean equalsWithRelativeTolerance(Complex x,
                                                       Complex y,
                                                       double eps) {
         return Precision.equalsWithRelativeTolerance(x.real, y.real, eps) &&
                Precision.equalsWithRelativeTolerance(x.imaginary, y.imaginary, eps);
     }
 
     /**
      * Get a hashCode for the complex number.
      * Any {@code Double.NaN} value in real or imaginary part produces
      * the same hash code {@code 7}.
      *
      * @return a hash code value for this object.
      */
     @Override
     public int hashCode() {
         if (isNaN) {
             return 7;
         }
         return 37 * (17 * MathUtils.hash(imaginary) +
             MathUtils.hash(real));
     }
 
     /** {@inheritDoc}
      * <p>
      * This implementation considers +0.0 and -0.0 to be equal for both
      * real and imaginary components.
      * </p>
      * @since 1.8
      */
     @Override
     public boolean isZero() {
         return real == 0.0 && imaginary == 0.0;
     }
 
     /**
      * Access the imaginary part.
      *
      * @return the imaginary part.
      */
     public double getImaginary() {
         return imaginary;
     }
 
     /**
      * Access the imaginary part.
      *
      * @return the imaginary part.
      * @since 2.0
      */
     public double getImaginaryPart() {
         return imaginary;
     }
 
     /**
      * Access the real part.
      *
      * @return the real part.
      */
     @Override
     public double getReal() {
         return real;
     }
 
     /**
      * Access the real part.
      *
      * @return the real part.
      * @since 2.0
      */
     public double getRealPart() {
         return real;
     }
 
     /**
      * Checks whether either or both parts of this complex number is
      * {@code NaN}.
      *
      * @return true if either or both parts of this complex number is
      * {@code NaN}; false otherwise.
      */
     @Override
     public boolean isNaN() {
         return isNaN;
     }
 
     /** Check whether the instance is real (i.e. imaginary part is zero).
      * @return true if imaginary part is zero
      * @since 1.7
      */
     public boolean isReal() {
         return imaginary == 0.0;
     }
 
     /** Check whether the instance is an integer (i.e. imaginary part is zero and real part has no fractional part).
      * @return true if imaginary part is zero and real part has no fractional part
      * @since 1.7
      */
     public boolean isMathematicalInteger() {
         return isReal() && Precision.isMathematicalInteger(real);
     }
 
     /**
      * Checks whether either the real or imaginary part of this complex number
      * takes an infinite value (either {@code Double.POSITIVE_INFINITY} or
      * {@code Double.NEGATIVE_INFINITY}) and neither part
      * is {@code NaN}.
      *
      * @return true if one or both parts of this complex number are infinite
      * and neither part is {@code NaN}.
      */
     @Override
     public boolean isInfinite() {
         return isInfinite;
     }
 
     /**
      * Returns a {@code Complex} whose value is {@code this * factor}.
      * Implements preliminary checks for {@code NaN} and infinity followed by
      * the definitional formula:
      * <p>
      *   {@code (a + bi)(c + di) = (ac - bd) + (ad + bc)i}
      * </p>
      * Returns {@link #NaN} if either {@code this} or {@code factor} has one or
      * more {@code NaN} parts.
      * <p>
      * Returns {@link #INF} if neither {@code this} nor {@code factor} has one
      * or more {@code NaN} parts and if either {@code this} or {@code factor}
      * has one or more infinite parts (same result is returned regardless of
      * the sign of the components).
      * </p><p>
      * Returns finite values in components of the result per the definitional
      * formula in all remaining cases.</p>
      *
      * @param  factor value to be multiplied by this {@code Complex}.
      * @return {@code this * factor}.
      * @throws NullArgumentException if {@code factor} is {@code null}.
      */
     @Override
     public Complex multiply(Complex factor)
         throws NullArgumentException {
         MathUtils.checkNotNull(factor);
         if (isNaN || factor.isNaN) {
             return NaN;
         }
         if (Double.isInfinite(real) ||
             Double.isInfinite(imaginary) ||
             Double.isInfinite(factor.real) ||
             Double.isInfinite(factor.imaginary)) {
             // we don't use isInfinite() to avoid testing for NaN again
             return INF;
         }
         return createComplex(MathArrays.linearCombination(real, factor.real, -imaginary, factor.imaginary),
                              MathArrays.linearCombination(real, factor.imaginary, imaginary, factor.real));
     }
 
     /**
      * Returns a {@code Complex} whose value is {@code this * factor}, with {@code factor}
      * interpreted as a integer number.
      *
      * @param  factor value to be multiplied by this {@code Complex}.
      * @return {@code this * factor}.
      * @see #multiply(Complex)
      */
     @Override
     public Complex multiply(final int factor) {
         if (isNaN) {
             return NaN;
         }
         if (Double.isInfinite(real) ||
             Double.isInfinite(imaginary)) {
             return INF;
         }
         return createComplex(real * factor, imaginary * factor);
     }
 
     /**
      * Returns a {@code Complex} whose value is {@code this * factor}, with {@code factor}
      * interpreted as a real number.
      *
      * @param  factor value to be multiplied by this {@code Complex}.
      * @return {@code this * factor}.
      * @see #multiply(Complex)
      */
     @Override
     public Complex multiply(double factor) {
         if (isNaN || Double.isNaN(factor)) {
             return NaN;
         }
         if (Double.isInfinite(real) ||
             Double.isInfinite(imaginary) ||
             Double.isInfinite(factor)) {
             // we don't use isInfinite() to avoid testing for NaN again
             return INF;
         }
         return createComplex(real * factor, imaginary * factor);
     }
 
     /** Compute this * i.
      * @return this * i
      * @since 2.0
      */
     public Complex multiplyPlusI() {
         return createComplex(-imaginary, real);
     }
 
     /** Compute this *- -i.
      * @return this * i
      * @since 2.0
      */
     public Complex multiplyMinusI() {
         return createComplex(imaginary, -real);
     }
 
     /** {@inheritDoc} */
     @Override
     public Complex square() {
         return multiply(this);
     }
 
     /**
      * Returns a {@code Complex} whose value is {@code (-this)}.
      * Returns {@code NaN} if either real or imaginary
      * part of this Complex number is {@code Double.NaN}.
      *
      * @return {@code -this}.
      */
     @Override
     public Complex negate() {
         if (isNaN) {
             return NaN;
         }
 
         return createComplex(-real, -imaginary);
     }
 
     /**
      * Returns a {@code Complex} whose value is
      * {@code (this - subtrahend)}.
      * Uses the definitional formula
      * <p>
      *  {@code (a + bi) - (c + di) = (a-c) + (b-d)i}
      * </p>
      * If either {@code this} or {@code subtrahend} has a {@code NaN]} value in either part,
      * {@link #NaN} is returned; otherwise infinite and {@code NaN} values are
      * returned in the parts of the result according to the rules for
      * {@link java.lang.Double} arithmetic.
      *
      * @param  subtrahend value to be subtracted from this {@code Complex}.
      * @return {@code this - subtrahend}.
      * @throws NullArgumentException if {@code subtrahend} is {@code null}.
      */
     @Override
     public Complex subtract(Complex subtrahend)
         throws NullArgumentException {
         MathUtils.checkNotNull(subtrahend);
         if (isNaN || subtrahend.isNaN) {
             return NaN;
         }
 
         return createComplex(real - subtrahend.getRealPart(),
                              imaginary - subtrahend.getImaginaryPart());
     }
 
     /**
      * Returns a {@code Complex} whose value is
      * {@code (this - subtrahend)}.
      *
      * @param  subtrahend value to be subtracted from this {@code Complex}.
      * @return {@code this - subtrahend}.
      * @see #subtract(Complex)
      */
     @Override
     public Complex subtract(double subtrahend) {
         if (isNaN || Double.isNaN(subtrahend)) {
             return NaN;
         }
         return createComplex(real - subtrahend, imaginary);
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/InverseCosine.html" TARGET="_top">
      * inverse cosine</a> of this complex number.
      * Implements the formula:
      * <p>
      *  {@code acos(z) = -i (log(z + i (sqrt(1 - z<sup>2</sup>))))}
      * </p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN} or infinite.
      *
      * @return the inverse cosine of this complex number.
      */
     @Override
     public Complex acos() {
         if (isNaN) {
             return NaN;
         }
 
         return this.add(this.sqrt1z().multiplyPlusI()).log().multiplyMinusI();
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/InverseSine.html" TARGET="_top">
      * inverse sine</a> of this complex number.
      * Implements the formula:
      * <p>
      *  {@code asin(z) = -i (log(sqrt(1 - z<sup>2</sup>) + iz))}
      * </p><p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN} or infinite.</p>
      *
      * @return the inverse sine of this complex number.
      */
     @Override
     public Complex asin() {
         if (isNaN) {
             return NaN;
         }
 
         return sqrt1z().add(this.multiplyPlusI()).log().multiplyMinusI();
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/InverseTangent.html" TARGET="_top">
      * inverse tangent</a> of this complex number.
      * Implements the formula:
      * <p>
      * {@code atan(z) = (i/2) log((1 - iz)/(1 + iz))}
      * </p><p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN} or infinite.</p>
      *
      * @return the inverse tangent of this complex number
      */
     @Override
     public Complex atan() {
         if (isNaN) {
             return NaN;
         }
 
         if (real == 0.0) {
 
             // singularity at ±i
             if (imaginary * imaginary - 1.0 == 0.0) {
                 return NaN;
             }
 
             // branch cut on imaginary axis
             final Complex tmp = createComplex((1 + imaginary) / (1 - imaginary), 0.0).log().multiplyPlusI().multiply(0.5);
             return createComplex(FastMath.copySign(tmp.real, real), tmp.imaginary);
 
         } else {
             // regular formula
             final Complex n = createComplex(1 + imaginary, -real);
             final Complex d = createComplex(1 - imaginary,  real);
             return n.divide(d).log().multiplyPlusI().multiply(0.5);
         }
 
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/Cosine.html" TARGET="_top">
      * cosine</a> of this complex number.
      * Implements the formula:
      * <p>
      *  {@code cos(a + bi) = cos(a)cosh(b) - sin(a)sinh(b)i}
      * </p><p>
      * where the (real) functions on the right-hand side are
      * {@link FastMath#sin}, {@link FastMath#cos},
      * {@link FastMath#cosh} and {@link FastMath#sinh}.
      * </p><p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p><p>
      * Infinite values in real or imaginary parts of the input may result in
      * infinite or NaN values returned in parts of the result.</p>
      * <pre>
      *  Examples:
      *  <code>
      *   cos(1 &plusmn; INFINITY i) = 1 \u2213 INFINITY i
      *   cos(&plusmn;INFINITY + i) = NaN + NaN i
      *   cos(&plusmn;INFINITY &plusmn; INFINITY i) = NaN + NaN i
      *  </code>
      * </pre>
      *
      * @return the cosine of this complex number.
      */
     @Override
     public Complex cos() {
         if (isNaN) {
             return NaN;
         }
 
         final SinCos   scr  = FastMath.sinCos(real);
         final SinhCosh schi = FastMath.sinhCosh(imaginary);
         return createComplex(scr.cos() * schi.cosh(), -scr.sin() * schi.sinh());
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/HyperbolicCosine.html" TARGET="_top">
      * hyperbolic cosine</a> of this complex number.
      * Implements the formula:
      * <pre>
      *  <code>
      *   cosh(a + bi) = cosh(a)cos(b) + sinh(a)sin(b)i
      *  </code>
      * </pre>
      * where the (real) functions on the right-hand side are
      * {@link FastMath#sin}, {@link FastMath#cos},
      * {@link FastMath#cosh} and {@link FastMath#sinh}.
      * <p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p>
      * Infinite values in real or imaginary parts of the input may result in
      * infinite or NaN values returned in parts of the result.
      * <pre>
      *  Examples:
      *  <code>
      *   cosh(1 &plusmn; INFINITY i) = NaN + NaN i
      *   cosh(&plusmn;INFINITY + i) = INFINITY &plusmn; INFINITY i
      *   cosh(&plusmn;INFINITY &plusmn; INFINITY i) = NaN + NaN i
      *  </code>
      * </pre>
      *
      * @return the hyperbolic cosine of this complex number.
      */
     @Override
     public Complex cosh() {
         if (isNaN) {
             return NaN;
         }
 
         final SinhCosh schr = FastMath.sinhCosh(real);
         final SinCos   sci  = FastMath.sinCos(imaginary);
         return createComplex(schr.cosh() * sci.cos(), schr.sinh() * sci.sin());
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/ExponentialFunction.html" TARGET="_top">
      * exponential function</a> of this complex number.
      * Implements the formula:
      * <pre>
      *  <code>
      *   exp(a + bi) = exp(a)cos(b) + exp(a)sin(b)i
      *  </code>
      * </pre>
      * where the (real) functions on the right-hand side are
      * {@link FastMath#exp(double)} p}, {@link FastMath#cos(double)}, and
      * {@link FastMath#sin(double)}.
      * <p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p>
      * Infinite values in real or imaginary parts of the input may result in
      * infinite or NaN values returned in parts of the result.
      * <pre>
      *  Examples:
      *  <code>
      *   exp(1 &plusmn; INFINITY i) = NaN + NaN i
      *   exp(INFINITY + i) = INFINITY + INFINITY i
      *   exp(-INFINITY + i) = 0 + 0i
      *   exp(&plusmn;INFINITY &plusmn; INFINITY i) = NaN + NaN i
      *  </code>
      * </pre>
      *
      * @return <code><i>e</i><sup>this</sup></code>.
      */
     @Override
     public Complex exp() {
         if (isNaN) {
             return NaN;
         }
 
         final double expReal = FastMath.exp(real);
         final SinCos sc      = FastMath.sinCos(imaginary);
         return createComplex(expReal * sc.cos(), expReal * sc.sin());
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex expm1() {
         if (isNaN) {
             return NaN;
         }
 
         final double expm1Real = FastMath.expm1(real);
         final SinCos sc        = FastMath.sinCos(imaginary);
         return createComplex(expm1Real * sc.cos(), expm1Real * sc.sin());
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/NaturalLogarithm.html" TARGET="_top">
      * natural logarithm</a> of this complex number.
      * Implements the formula:
      * <pre>
      *  <code>
      *   log(a + bi) = ln(|a + bi|) + arg(a + bi)i
      *  </code>
      * </pre>
      * where ln on the right hand side is {@link FastMath#log(double)},
      * {@code |a + bi|} is the modulus, {@link Complex#abs},  and
      * {@code arg(a + bi) = }{@link FastMath#atan2}(b, a).
      * <p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p>
      * 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.
      * <pre>
      *  Examples:
      *  <code>
      *   log(1 &plusmn; INFINITY i) = INFINITY &plusmn; (&pi;/2)i
      *   log(INFINITY + i) = INFINITY + 0i
      *   log(-INFINITY + i) = INFINITY + &pi;i
      *   log(INFINITY &plusmn; INFINITY i) = INFINITY &plusmn; (&pi;/4)i
      *   log(-INFINITY &plusmn; INFINITY i) = INFINITY &plusmn; (3&pi;/4)i
      *   log(0 + 0i) = -INFINITY + 0i
      *  </code>
      * </pre>
      *
      * @return the value <code>ln &nbsp; this</code>, the natural logarithm
      * of {@code this}.
      */
     @Override
     public Complex log() {
         if (isNaN) {
             return NaN;
         }
 
         return createComplex(FastMath.log(FastMath.hypot(real, imaginary)),
                              FastMath.atan2(imaginary, real));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex log1p() {
         return add(1.0).log();
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex log10() {
         return log().divide(LOG10);
     }
 
     /**
      * Returns of value of this complex number raised to the power of {@code x}.
      * <p>
      * If {@code x} is a real number whose real part has an integer value, returns {@link #pow(int)},
      * if both {@code this} and {@code x} are real and {@link FastMath#pow(double, double)}
      * with the corresponding real arguments would return a finite number (neither NaN
      * nor infinite), then returns the same value converted to {@code Complex},
      * with the same special cases.
      * In all other cases real cases, implements y<sup>x</sup> = exp(x&middot;log(y)).
      * </p>
      *
      * @param  x exponent to which this {@code Complex} is to be raised.
      * @return <code> this<sup>x</sup></code>.
      * @throws NullArgumentException if x is {@code null}.
      */
     @Override
     public Complex pow(Complex x)
         throws NullArgumentException {
 
         MathUtils.checkNotNull(x);
 
         if (x.imaginary == 0.0) {
             final int nx = (int) FastMath.rint(x.real);
             if (x.real == nx) {
                 // integer power
                 return pow(nx);
             } else if (this.imaginary == 0.0) {
                 // check real implementation that handles a bunch of special cases
                 final double realPow = FastMath.pow(this.real, x.real);
                 if (Double.isFinite(realPow)) {
                     return createComplex(realPow, 0);
                 }
             }
         }
 
         // generic implementation
         return this.log().multiply(x).exp();
 
     }
 
 
     /**
      * Returns of value of this complex number raised to the power of {@code x}.
      * <p>
      * If {@code x} has an integer value, returns {@link #pow(int)},
      * if {@code this} is real and {@link FastMath#pow(double, double)}
      * with the corresponding real arguments would return a finite number (neither NaN
      * nor infinite), then returns the same value converted to {@code Complex},
      * with the same special cases.
      * In all other cases real cases, implements y<sup>x</sup> = exp(x&middot;log(y)).
      * </p>
      *
      * @param  x exponent to which this {@code Complex} is to be raised.
      * @return <code> this<sup>x</sup></code>.
      */
     @Override
     public Complex pow(double x) {
 
         final int nx = (int) FastMath.rint(x);
         if (x == nx) {
             // integer power
             return pow(nx);
         } else if (this.imaginary == 0.0) {
             // check real implementation that handles a bunch of special cases
             final double realPow = FastMath.pow(this.real, x);
             if (Double.isFinite(realPow)) {
                 return createComplex(realPow, 0);
             }
         }
 
         // generic implementation
         return this.log().multiply(x).exp();
 
     }
 
      /** {@inheritDoc}
       * @since 1.7
       */
     @Override
     public Complex pow(final int n) {
 
         Complex result = ONE;
         final boolean invert;
         int p = n;
         if (p < 0) {
             invert = true;
             p = -p;
         } else {
             invert = false;
         }
 
         // Exponentiate by successive squaring
         Complex square = this;
         while (p > 0) {
             if ((p & 0x1) > 0) {
                 result = result.multiply(square);
             }
             square = square.multiply(square);
             p = p >> 1;
         }
 
         return invert ? result.reciprocal() : result;
 
     }
 
      /**
       * Compute the
      * <a href="http://mathworld.wolfram.com/Sine.html" TARGET="_top">
      * sine</a>
      * of this complex number.
      * Implements the formula:
      * <pre>
      *  <code>
      *   sin(a + bi) = sin(a)cosh(b) + cos(a)sinh(b)i
      *  </code>
      * </pre>
      * where the (real) functions on the right-hand side are
      * {@link FastMath#sin}, {@link FastMath#cos},
      * {@link FastMath#cosh} and {@link FastMath#sinh}.
      * <p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p><p>
      * Infinite values in real or imaginary parts of the input may result in
      * infinite or {@code NaN} values returned in parts of the result.
      * <pre>
      *  Examples:
      *  <code>
      *   sin(1 &plusmn; INFINITY i) = 1 &plusmn; INFINITY i
      *   sin(&plusmn;INFINITY + i) = NaN + NaN i
      *   sin(&plusmn;INFINITY &plusmn; INFINITY i) = NaN + NaN i
      *  </code>
      * </pre>
      *
      * @return the sine of this complex number.
      */
     @Override
     public Complex sin() {
         if (isNaN) {
             return NaN;
         }
 
         final SinCos   scr  = FastMath.sinCos(real);
         final SinhCosh schi = FastMath.sinhCosh(imaginary);
         return createComplex(scr.sin() * schi.cosh(), scr.cos() * schi.sinh());
 
     }
 
     /** {@inheritDoc}
      */
     @Override
     public FieldSinCos<Complex> sinCos() {
         if (isNaN) {
             return new FieldSinCos<>(NaN, NaN);
         }
 
         final SinCos scr = FastMath.sinCos(real);
         final SinhCosh schi = FastMath.sinhCosh(imaginary);
         return new FieldSinCos<>(createComplex(scr.sin() * schi.cosh(),  scr.cos() * schi.sinh()),
                                  createComplex(scr.cos() * schi.cosh(), -scr.sin() * schi.sinh()));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex atan2(Complex x) {
 
         // compute r = sqrt(x^2+y^2)
         final Complex r = x.square().add(multiply(this)).sqrt();
 
         if (FastMath.copySign(1.0, x.real) >= 0) {
             // compute atan2(y, x) = 2 atan(y / (r + x))
             return divide(r.add(x)).atan().multiply(2);
         } else {
             // compute atan2(y, x) = +/- pi - 2 atan(y / (r - x))
             return divide(r.subtract(x)).atan().multiply(-2).add(FastMath.PI);
         }
     }
 
     /** {@inheritDoc}
      * <p>
      * Branch cuts are on the real axis, below +1.
      * </p>
      * @since 1.7
      */
     @Override
     public Complex acosh() {
         final Complex sqrtPlus  = add(1).sqrt();
         final Complex sqrtMinus = subtract(1).sqrt();
         return add(sqrtPlus.multiply(sqrtMinus)).log();
     }
 
     /** {@inheritDoc}
      * <p>
      * Branch cuts are on the imaginary axis, above +i and below -i.
      * </p>
      * @since 1.7
      */
     @Override
     public Complex asinh() {
         return add(multiply(this).add(1.0).sqrt()).log();
     }
 
     /** {@inheritDoc}
      * <p>
      * Branch cuts are on the real axis, above +1 and below -1.
      * </p>
      * @since 1.7
      */
     @Override
     public Complex atanh() {
         final Complex logPlus  = add(1).log();
         final Complex logMinus = createComplex(1 - real, -imaginary).log();
         return logPlus.subtract(logMinus).multiply(0.5);
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/HyperbolicSine.html" TARGET="_top">
      * hyperbolic sine</a> of this complex number.
      * Implements the formula:
      * <pre>
      *  <code>
      *   sinh(a + bi) = sinh(a)cos(b)) + cosh(a)sin(b)i
      *  </code>
      * </pre>
      * where the (real) functions on the right-hand side are
      * {@link FastMath#sin}, {@link FastMath#cos},
      * {@link FastMath#cosh} and {@link FastMath#sinh}.
      * <p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p><p>
      * Infinite values in real or imaginary parts of the input may result in
      * infinite or NaN values returned in parts of the result.
      * <pre>
      *  Examples:
      *  <code>
      *   sinh(1 &plusmn; INFINITY i) = NaN + NaN i
      *   sinh(&plusmn;INFINITY + i) = &plusmn; INFINITY + INFINITY i
      *   sinh(&plusmn;INFINITY &plusmn; INFINITY i) = NaN + NaN i
      *  </code>
      * </pre>
      *
      * @return the hyperbolic sine of {@code this}.
      */
     @Override
     public Complex sinh() {
         if (isNaN) {
             return NaN;
         }
 
         final SinhCosh schr = FastMath.sinhCosh(real);
         final SinCos   sci  = FastMath.sinCos(imaginary);
         return createComplex(schr.sinh() * sci.cos(), schr.cosh() * sci.sin());
     }
 
     /** {@inheritDoc}
      */
     @Override
     public FieldSinhCosh<Complex> sinhCosh() {
         if (isNaN) {
             return new FieldSinhCosh<>(NaN, NaN);
         }
 
         final SinhCosh schr = FastMath.sinhCosh(real);
         final SinCos   sci  = FastMath.sinCos(imaginary);
         return new FieldSinhCosh<>(createComplex(schr.sinh() * sci.cos(), schr.cosh() * sci.sin()),
                                    createComplex(schr.cosh() * sci.cos(), schr.sinh() * sci.sin()));
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/SquareRoot.html" TARGET="_top">
      * square root</a> of this complex number.
      * Implements the following algorithm to compute {@code sqrt(a + bi)}:
      * <ol><li>Let {@code t = sqrt((|a| + |a + bi|) / 2)}</li>
      * <li><pre>if {@code  a ≥ 0} return {@code t + (b/2t)i}
      *  else return {@code |b|/2t + sign(b)t i }</pre></li>
      * </ol>
      * where <ul>
      * <li>{@code |a| = }{@link FastMath#abs(double) abs(a)}</li>
      * <li>{@code |a + bi| = }{@link FastMath#hypot(double, double) hypot(a, b)}</li>
      * <li>{@code sign(b) = }{@link FastMath#copySign(double, double) copySign(1, b)}
      * </ul>
      * The real part is therefore always nonnegative.
      * <p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p>
      * <p>
      * Infinite values in real or imaginary parts of the input may result in
      * infinite or NaN values returned in parts of the result.
      * </p>
      * <pre>
      *  Examples:
      *  <code>
      *   sqrt(1 ± ∞ i) = ∞ + NaN i
      *   sqrt(∞ + i) = ∞ + 0i
      *   sqrt(-∞ + i) = 0 + ∞ i
      *   sqrt(∞ ± ∞ i) = ∞ + NaN i
      *   sqrt(-∞ ± ∞ i) = NaN ± ∞ i
      *  </code>
      * </pre>
      *
      * @return the square root of {@code this} with nonnegative real part.
      */
     @Override
     public Complex sqrt() {
         if (isNaN) {
             return NaN;
         }
 
         if (real == 0.0 && imaginary == 0.0) {
             return ZERO;
         }
 
         double t = FastMath.sqrt((FastMath.abs(real) + FastMath.hypot(real, imaginary)) * 0.5);
         if (FastMath.copySign(1, real) >= 0.0) {
             return createComplex(t, imaginary / (2.0 * t));
         } else {
             return createComplex(FastMath.abs(imaginary) / (2.0 * t),
                                  FastMath.copySign(t, imaginary));
         }
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/SquareRoot.html" TARGET="_top">
      * square root</a> of <code>1 - this<sup>2</sup></code> for this complex
      * number.
      * Computes the result directly as
      * {@code sqrt(ONE.subtract(z.square()))}.
      * <p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p>
      * Infinite values in real or imaginary parts of the input may result in
      * infinite or NaN values returned in parts of the result.
      *
      * @return the square root of <code>1 - this<sup>2</sup></code>.
      */
     public Complex sqrt1z() {
         final Complex t2 = this.square();
         return createComplex(1 - t2.real, -t2.imaginary).sqrt();
     }
 
     /** {@inheritDoc}
      * <p>
      * This implementation compute the principal cube root by using a branch cut along real negative axis.
      * </p>
      * @since 1.7
      */
     @Override
     public Complex cbrt() {
         final double magnitude = FastMath.cbrt(norm());
         final SinCos sc        = FastMath.sinCos(getArgument() / 3);
         return createComplex(magnitude * sc.cos(), magnitude * sc.sin());
     }
 
     /** {@inheritDoc}
      * <p>
      * This implementation compute the principal n<sup>th</sup> root by using a branch cut along real negative axis.
      * </p>
      * @since 1.7
      */
     @Override
     public Complex rootN(int n) {
         final double magnitude = FastMath.pow(norm(), 1.0 / n);
         final SinCos sc        = FastMath.sinCos(getArgument() / n);
         return createComplex(magnitude * sc.cos(), magnitude * sc.sin());
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/Tangent.html" TARGET="_top">
      * tangent</a> of this complex number.
      * Implements the formula:
      * <pre>
      *  <code>
      *   tan(a + bi) = sin(2a)/(cos(2a)+cosh(2b)) + [sinh(2b)/(cos(2a)+cosh(2b))]i
      *  </code>
      * </pre>
      * where the (real) functions on the right-hand side are
      * {@link FastMath#sin}, {@link FastMath#cos}, {@link FastMath#cosh} and
      * {@link FastMath#sinh}.
      * <p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p>
      * 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.
      * <pre>
      *  Examples:
      *  <code>
      *   tan(a &plusmn; INFINITY i) = 0 &plusmn; i
      *   tan(&plusmn;INFINITY + bi) = NaN + NaN i
      *   tan(&plusmn;INFINITY &plusmn; INFINITY i) = NaN + NaN i
      *   tan(&plusmn;&pi;/2 + 0 i) = &plusmn;INFINITY + NaN i
      *  </code>
      * </pre>
      *
      * @return the tangent of {@code this}.
      */
     @Override
     public Complex tan() {
         if (isNaN || Double.isInfinite(real)) {
             return NaN;
         }
         if (imaginary > 20.0) {
             return I;
         }
         if (imaginary < -20.0) {
             return MINUS_I;
         }
 
         final SinCos sc2r = FastMath.sinCos(2.0 * real);
         double imaginary2 = 2.0 * imaginary;
         double d = sc2r.cos() + FastMath.cosh(imaginary2);
 
         return createComplex(sc2r.sin() / d, FastMath.sinh(imaginary2) / d);
 
     }
 
     /**
      * Compute the
      * <a href="http://mathworld.wolfram.com/HyperbolicTangent.html" TARGET="_top">
      * hyperbolic tangent</a> of this complex number.
      * Implements the formula:
      * <pre>
      *  <code>
      *   tan(a + bi) = sinh(2a)/(cosh(2a)+cos(2b)) + [sin(2b)/(cosh(2a)+cos(2b))]i
      *  </code>
      * </pre>
      * where the (real) functions on the right-hand side are
      * {@link FastMath#sin}, {@link FastMath#cos}, {@link FastMath#cosh} and
      * {@link FastMath#sinh}.
      * <p>
      * Returns {@link Complex#NaN} if either real or imaginary part of the
      * input argument is {@code NaN}.
      * </p>
      * Infinite values in real or imaginary parts of the input may result in
      * infinite or NaN values returned in parts of the result.
      * <pre>
      *  Examples:
      *  <code>
      *   tanh(a &plusmn; INFINITY i) = NaN + NaN i
      *   tanh(&plusmn;INFINITY + bi) = &plusmn;1 + 0 i
      *   tanh(&plusmn;INFINITY &plusmn; INFINITY i) = NaN + NaN i
      *   tanh(0 + (&pi;/2)i) = NaN + INFINITY i
      *  </code>
      * </pre>
      *
      * @return the hyperbolic tangent of {@code this}.
      */
     @Override
     public Complex tanh() {
         if (isNaN || Double.isInfinite(imaginary)) {
             return NaN;
         }
         if (real > 20.0) {
             return ONE;
         }
         if (real < -20.0) {
             return MINUS_ONE;
         }
         double real2 = 2.0 * real;
         final SinCos sc2i = FastMath.sinCos(2.0 * imaginary);
         double d = FastMath.cosh(real2) + sc2i.cos();
 
         return createComplex(FastMath.sinh(real2) / d, sc2i.sin() / d);
     }
 
 
 
     /**
      * 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.
      * <p>
      * If either real or imaginary part (or both) is NaN, NaN is returned.
      * Infinite parts are handled as {@code 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 {@code Math.atan2} for full details.
      *
      * @return the argument of {@code this}.
      */
     public double getArgument() {
         return FastMath.atan2(getImaginaryPart(), getRealPart());
     }
 
     /**
      * Computes the n-th roots of this complex number.
      * The nth roots are defined by the formula:
      * <pre>
      *  <code>
      *   z<sub>k</sub> = abs<sup>1/n</sup> (cos(phi + 2&pi;k/n) + i (sin(phi + 2&pi;k/n))
      *  </code>
      * </pre>
      * for <i>{@code k=0, 1, ..., n-1}</i>, where {@code abs} and {@code phi}
      * are respectively the {@link #abs() modulus} and
      * {@link #getArgument() argument} of this complex number.
      * <p>
      * If one or both parts of this complex number is NaN, a list with just
      * one element, {@link #NaN} is returned.
      * if neither part is NaN, but at least one part is infinite, the result
      * is a one-element list containing {@link #INF}.
      *
      * @param n Degree of root.
      * @return a List of all {@code n}-th roots of {@code this}.
      * @throws MathIllegalArgumentException if {@code n <= 0}.
      */
     public List<Complex> nthRoot(int n) throws MathIllegalArgumentException {
 
         if (n <= 0) {
             throw new MathIllegalArgumentException(LocalizedCoreFormats.CANNOT_COMPUTE_NTH_ROOT_FOR_NEGATIVE_N,
                                                    n);
         }
 
         final List<Complex> result = new ArrayList<>();
 
         if (isNaN) {
             result.add(NaN);
             return result;
         }
         if (isInfinite()) {
             result.add(INF);
             return result;
         }
 
         // nth root of abs -- faster / more accurate to use a solver here?
         final double nthRootOfAbs = FastMath.pow(FastMath.hypot(real, imaginary), 1.0 / n);
 
         // Compute nth roots of complex number with k = 0, 1, ... n-1
         final double nthPhi = getArgument() / n;
         final double slice = 2 * FastMath.PI / n;
         double innerPart = nthPhi;
         for (int k = 0; k < n ; k++) {
             // inner part
             final SinCos scInner = FastMath.sinCos(innerPart);
             final double realPart = nthRootOfAbs *  scInner.cos();
             final double imaginaryPart = nthRootOfAbs *  scInner.sin();
             result.add(createComplex(realPart, imaginaryPart));
             innerPart += slice;
         }
 
         return result;
     }
 
     /**
      * Create a complex number given the real and imaginary parts.
      *
      * @param realPart Real part.
      * @param imaginaryPart Imaginary part.
      * @return a new complex number instance.
      *
      * @see #valueOf(double, double)
      */
     protected Complex createComplex(double realPart,
                                     double imaginaryPart) {
         return new Complex(realPart, imaginaryPart);
     }
 
     /**
      * Create a complex number given the real and imaginary parts.
      *
      * @param realPart Real part.
      * @param imaginaryPart Imaginary part.
      * @return a Complex instance.
      */
     public static Complex valueOf(double realPart,
                                   double imaginaryPart) {
         if (Double.isNaN(realPart) ||
             Double.isNaN(imaginaryPart)) {
             return NaN;
         }
         return new Complex(realPart, imaginaryPart);
     }
 
     /**
      * Create a complex number given only the real part.
      *
      * @param realPart Real part.
      * @return a Complex instance.
      */
     public static Complex valueOf(double realPart) {
         if (Double.isNaN(realPart)) {
             return NaN;
         }
         return new Complex(realPart);
     }
 
     /** {@inheritDoc} */
     @Override
     public Complex newInstance(double realPart) {
         return valueOf(realPart);
     }
 
     /**
      * Resolve the transient fields in a deserialized Complex Object.
      * Subclasses will need to override {@link #createComplex} to
      * deserialize properly.
      *
      * @return A Complex instance with all fields resolved.
      */
     protected final Object readResolve() {
         return createComplex(real, imaginary);
     }
 
     /** {@inheritDoc} */
     @Override
     public ComplexField getField() {
         return ComplexField.getInstance();
     }
 
     /** {@inheritDoc} */
     @Override
     public String toString() {
         return "(" + real + ", " + imaginary + ")";
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex scalb(int n) {
         return createComplex(FastMath.scalb(real, n), FastMath.scalb(imaginary, n));
     }
 
     /** {@inheritDoc}
      */
     @Override
     public Complex ulp() {
         return createComplex(FastMath.ulp(real), FastMath.ulp(imaginary));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex hypot(Complex y) {
         if (isInfinite() || y.isInfinite()) {
             return INF;
         } else if (isNaN() || y.isNaN()) {
             return NaN;
         } else {
             return square().add(y.square()).sqrt();
         }
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex linearCombination(final Complex[] a, final Complex[] b)
         throws MathIllegalArgumentException {
         final int n = 2 * a.length;
         final double[] realA      = new double[n];
         final double[] realB      = new double[n];
         final double[] imaginaryA = new double[n];
         final double[] imaginaryB = new double[n];
         for (int i = 0; i < a.length; ++i)  {
             final Complex ai = a[i];
             final Complex bi = b[i];
             realA[2 * i    ]      = +ai.real;
             realA[2 * i + 1]      = -ai.imaginary;
             realB[2 * i    ]      = +bi.real;
             realB[2 * i + 1]      = +bi.imaginary;
             imaginaryA[2 * i    ] = +ai.real;
             imaginaryA[2 * i + 1] = +ai.imaginary;
             imaginaryB[2 * i    ] = +bi.imaginary;
             imaginaryB[2 * i + 1] = +bi.real;
         }
         return createComplex(MathArrays.linearCombination(realA,  realB),
                              MathArrays.linearCombination(imaginaryA, imaginaryB));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex linearCombination(final double[] a, final Complex[] b)
         throws MathIllegalArgumentException {
         final int n = a.length;
         final double[] realB      = new double[n];
         final double[] imaginaryB = new double[n];
         for (int i = 0; i < a.length; ++i)  {
             final Complex bi = b[i];
             realB[i]      = +bi.real;
             imaginaryB[i] = +bi.imaginary;
         }
         return createComplex(MathArrays.linearCombination(a,  realB),
                              MathArrays.linearCombination(a, imaginaryB));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex linearCombination(final Complex a1, final Complex b1, final Complex a2, final Complex b2) {
         return createComplex(MathArrays.linearCombination(+a1.real, b1.real,
                                                           -a1.imaginary, b1.imaginary,
                                                           +a2.real, b2.real,
                                                           -a2.imaginary, b2.imaginary),
                              MathArrays.linearCombination(+a1.real, b1.imaginary,
                                                           +a1.imaginary, b1.real,
                                                           +a2.real, b2.imaginary,
                                                           +a2.imaginary, b2.real));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex linearCombination(final double a1, final Complex b1, final double a2, final Complex b2) {
         return createComplex(MathArrays.linearCombination(a1, b1.real,
                                                           a2, b2.real),
                              MathArrays.linearCombination(a1, b1.imaginary,
                                                           a2, b2.imaginary));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex linearCombination(final Complex a1, final Complex b1,
                                      final Complex a2, final Complex b2,
                                      final Complex a3, final Complex b3) {
         return linearCombination(new Complex[] { a1, a2, a3 },
                                  new Complex[] { b1, b2, b3 });
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex linearCombination(final double a1, final Complex b1,
                                      final double a2, final Complex b2,
                                      final double a3, final Complex b3) {
         return linearCombination(new double[]  { a1, a2, a3 },
                                  new Complex[] { b1, b2, b3 });
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex linearCombination(final Complex a1, final Complex b1,
                                      final Complex a2, final Complex b2,
                                      final Complex a3, final Complex b3,
                                      final Complex a4, final Complex b4) {
         return linearCombination(new Complex[] { a1, a2, a3, a4 },
                                  new Complex[] { b1, b2, b3, b4 });
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex linearCombination(final double a1, final Complex b1,
                                      final double a2, final Complex b2,
                                      final double a3, final Complex b3,
                                      final double a4, final Complex b4) {
         return linearCombination(new double[]  { a1, a2, a3, a4 },
                                  new Complex[] { b1, b2, b3, b4 });
     }
 
     /** {@inheritDoc} */
     @Override
     public Complex getPi() {
         return PI;
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex ceil() {
         return createComplex(FastMath.ceil(getRealPart()), FastMath.ceil(getImaginaryPart()));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex floor() {
         return createComplex(FastMath.floor(getRealPart()), FastMath.floor(getImaginaryPart()));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex rint() {
         return createComplex(FastMath.rint(getRealPart()), FastMath.rint(getImaginaryPart()));
     }
 
     /** {@inheritDoc}
      * <p>
      * for complex numbers, the integer n corresponding to {@code this.subtract(remainder(a)).divide(a)}
      * is a <a href="https://en.wikipedia.org/wiki/Gaussian_integer">Wikipedia - Gaussian integer</a>.
      * </p>
      * @since 1.7
      */
     @Override
     public Complex remainder(final double a) {
         return createComplex(FastMath.IEEEremainder(getRealPart(), a), FastMath.IEEEremainder(getImaginaryPart(), a));
     }
 
     /** {@inheritDoc}
      * <p>
      * for complex numbers, the integer n corresponding to {@code this.subtract(remainder(a)).divide(a)}
      * is a <a href="https://en.wikipedia.org/wiki/Gaussian_integer">Wikipedia - Gaussian integer</a>.
      * </p>
      * @since 1.7
      */
     @Override
     public Complex remainder(final Complex a) {
         final Complex complexQuotient = divide(a);
         final double  qRInt           = FastMath.rint(complexQuotient.real);
         final double  qIInt           = FastMath.rint(complexQuotient.imaginary);
         return createComplex(real - qRInt * a.real + qIInt * a.imaginary,
                              imaginary - qRInt * a.imaginary - qIInt * a.real);
     }
 
     /** {@inheritDoc}
      * @since 2.0
      */
     @Override
     public Complex sign() {
         if (isNaN() || isZero()) {
             return this;
         } else {
             return this.divide(FastMath.hypot(real, imaginary));
         }
     }
 
     /** {@inheritDoc}
      * <p>
      * The signs of real and imaginary parts are copied independently.
      * </p>
      * @since 1.7
      */
     @Override
     public Complex copySign(final Complex z) {
         return createComplex(FastMath.copySign(getRealPart(), z.getRealPart()),
                              FastMath.copySign(getImaginaryPart(), z.getImaginaryPart()));
     }
 
     /** {@inheritDoc}
      * @since 1.7
      */
     @Override
     public Complex copySign(double r) {
         return createComplex(FastMath.copySign(getRealPart(), r), FastMath.copySign(getImaginaryPart(), r));
     }
 
     /** {@inheritDoc} */
     @Override
     public Complex toDegrees() {
         return createComplex(FastMath.toDegrees(getRealPart()), FastMath.toDegrees(getImaginaryPart()));
     }
 
     /** {@inheritDoc} */
     @Override
     public Complex toRadians() {
         return createComplex(FastMath.toRadians(getRealPart()), FastMath.toRadians(getImaginaryPart()));
     }
 
     /** {@inheritDoc}
      * <p>
      * Comparison us performed using real ordering as the primary sort order and
      * imaginary ordering as the secondary sort order.
      * </p>
      * @since 3.0
      */
     @Override
     public int compareTo(final Complex o) {
         final int cR = Double.compare(getReal(), o.getReal());
         if (cR == 0) {
             return Double.compare(getImaginary(),o.getImaginary());
         } else {
             return cR;
         }
     }
 
 }