
public class Complex {

  private final double re;   // the real part
  private final double im;   // the imaginary part

  // create a new object with the given real and imaginary parts
  public Complex(double real, double imag) {
    re = real;
    im = imag;
  }

  // return a string representation of the invoking Complex object
  public String toString() {
    if (im == 0) return re + "";
    if (re == 0) return im + "i";
    if (im <  0) return re + " - " + (-im) + "i";
    return re + " + " + im + "i";
  }

  // return abs/modulus/magnitude and angle/phase/argument
  public double abs() { 
    return Math.hypot(re, im);
  }  // Math.sqrt(re*re + im*im)
  public double phase() { 
    return Math.atan2(im, re);
  }  // between -pi and pi

  // return a new Complex object whose value is (this + b)
  public Complex plus(Complex b) {
    Complex a = this;             // invoking object
    double real = a.re + b.re;
    double imag = a.im + b.im;
    return new Complex(real, imag);
  }

  // return a new Complex object whose value is (this - b)
  public Complex minus(Complex b) {
    Complex a = this;
    double real = a.re - b.re;
    double imag = a.im - b.im;
    return new Complex(real, imag);
  }

  // return a new Complex object whose value is (this * b)
  public Complex times(Complex b) {
    Complex a = this;
    double real = a.re * b.re - a.im * b.im;
    double imag = a.re * b.im + a.im * b.re;
    return new Complex(real, imag);
  }

  // scalar multiplication
  // return a new object whose value is (this * alpha)
  public Complex times(double alpha) {
    return new Complex(alpha * re, alpha * im);
  }

  // return a new Complex object whose value is the conjugate of this
  public Complex conjugate() {  
    return new Complex(re, -im);
  }

  // return a new Complex object whose value is the reciprocal of this
  public Complex reciprocal() {
    double scale = re*re + im*im;
    return new Complex(re / scale, -im / scale);
  }

  // return the real or imaginary part
  public double re() { 
    return re;
  }
  public double im() { 
    return im;
  }

  // return a / b
  public Complex divides(Complex b) {
    Complex a = this;
    return a.times(b.reciprocal());
  }

  // return a new Complex object whose value is the complex exponential of this
  public Complex exp() {
    return new Complex(Math.exp(re) * Math.cos(im), Math.exp(re) * Math.sin(im));
  }

  // return a new Complex object whose value is the complex sine of this
  public Complex sin() {
    return new Complex(Math.sin(re) * Math.cosh(im), Math.cos(re) * Math.sinh(im));
  }

  // return a new Complex object whose value is the complex cosine of this
  public Complex cos() {
    return new Complex(Math.cos(re) * Math.cosh(im), -Math.sin(re) * Math.sinh(im));
  }

  // return a new Complex object whose value is the complex tangent of this
  public Complex tan() {
    return sin().divides(cos());
  }



  // a static version of plus
  public static Complex plus(Complex a, Complex b) {
    double real = a.re + b.re;
    double imag = a.im + b.im;
    Complex sum = new Complex(real, imag);
    return sum;
  }



  // sample client for testing
  public static void main(String[] args) {
    Complex a = new Complex(5.0, 6.0);
    Complex b = new Complex(-3.0, 4.0);

    System.out.println("a            = " + a);
    System.out.println("b            = " + b);
    System.out.println("Re(a)        = " + a.re());
    System.out.println("Im(a)        = " + a.im());
    System.out.println("b + a        = " + b.plus(a));
    System.out.println("a - b        = " + a.minus(b));
    System.out.println("a * b        = " + a.times(b));
    System.out.println("b * a        = " + b.times(a));
    System.out.println("a / b        = " + a.divides(b));
    System.out.println("(a / b) * b  = " + a.divides(b).times(b));
    System.out.println("conj(a)      = " + a.conjugate());
    System.out.println("|a|          = " + a.abs());
    System.out.println("tan(a)       = " + a.tan());
  }
}
