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  <properties>
    <title>The Commons Math User Guide - Complex Numbers</title>
  </properties>
  <body>
    <section name="7 Complex Numbers">
      <subsection name="7.1 Overview" href="overview">
        <p>
          The complex packages provides a complex number type as well as complex
          versions of common transcendental functions and complex number
          formatting.
        </p>
      </subsection>
      <subsection name="7.2 Complex Numbers" href="complex">
        <p>
          <a href="../apidocs/org/apache/commons/math/complex/Complex.html">
          Complex</a> provides a complex number type that forms the basis for
          the complex functionality found in commons-math.
         </p>  
         <p>
           Complex functions and arithmetic operations are implemented in
           commons-math by applying standard computational formulas and
           following the rules for <code>java.lang.Double</code> arithmetic in 
           handling infinite and <code>NaN</code> values.  No attempt is made
           to comply with ANSII/IEC C99x Annex G or any other standard for
           Complex arithmetic.  See the class and method javadocs for the 
           <a href="../apidocs/org/apache/commons/math/complex/Complex.html">
           Complex</a> and
           <a href="../apidocs/org/apache/commons/math/complex/ComplexUtils.html">
           ComplexUtils</a> classes for details on computing formulas.
        </p>
        <p>
          To create a complex number, simply call the constructor passing in two
          floating-point arguments, the first being the real part of the
          complex number and the second being the imaginary part:
          <source>Complex c = new Complex(1.0, 3.0); // 1 + 3i</source>
        </p>
        <p>
          Complex numbers may also be created from polar representations
          using the <code>polar2Complex</code> method in 
          <code>ComplexUtils</code>.
        </p>
        <p>
          The <code>Complex</code> class provides basic unary and binary
          complex number operations.  These operations provide the means to add,
          subtract, multiply and divide complex numbers along with other
          complex number functions similar to the real number functions found in
          <code>java.math.BigDecimal</code>:
          <source>Complex lhs = new Complex(1.0, 3.0);
Complex rhs = new Complex(2.0, 5.0);

Complex answer = lhs.add(rhs);       // add two complex numbers
        answer = lhs.subtract(rhs);  // subtract two complex numbers
        answer = lhs.abs();          // absolute value
        answer = lhs.conjugate(rhs); // complex conjugate</source>
        </p>
      </subsection>
      <subsection name="7.3 Complex Transcendental Functions" href="function">
        <p>
          <a href="../apidocs/org/apache/commons/math/complex/Complex.html">
          Complex</a> also provides implementations of serveral transcendental
          functions involving complex number arguments.
          Prior to version 1.2, these functions were provided
          by <a href="../apidocs/org/apache/commons/math/complex/ComplexUtils.html">
          ComplexUtils</a> in a way similar to the real number functions found in
          <code>java.lang.Math</code>, but this has been deprecated.
          These operations provide the means to compute the log, sine, tangent,
          and other complex values :
          <source>Complex first  = new Complex(1.0, 3.0);
Complex second = new Complex(2.0, 5.0);

Complex answer = first.log();        // natural logarithm.
        answer = first.cos();        // cosine
        answer = first.pow(second);  // first raised to the power of second</source>
        </p>
      </subsection>
      <subsection name="7.4 Complex Formatting and Parsing" href="formatting">
        <p>
          <code>Complex</code> instances can be converted to and from strings
          using the<a href="../apidocs/org/apache/commons/math/complex/ComplexFormat.html">
          ComplexFormat</a> class.
          <code>ComplexFormat</code> is a <code>java.text.Format</code>
          extension and, as such, is used like other formatting objects (e.g.
          <code>java.text.SimpleDateFormat</code>):
          <source>ComplexFormat format = new ComplexFormat(); // default format
Complex c = new Complex(1.1111, 2.2222);
String s = format.format(c); // s contains "1.11 + 2.22i"</source>
        </p>
        <p>
          To customize the formatting output, one or two
          <code>java.text.NumberFormat</code> instances can be used to construct
          a <code>ComplexFormat</code>.  These number formats control the
          formatting of the real and imaginary values of the complex number:
          <source>NumberFormat nf = NumberFormat.getInstance();
nf.setMinimumFractionDigits(3);
nf.setMaximumFractionDigits(3);

// create complex format with custom number format
// when one number format is used, both real and
// imaginary parts are formatted the same
ComplexFormat cf = new ComplexFormat(nf);
Complex c = new Complex(1.11, 2.2222);
String s = format.format(c); // s contains "1.110 + 2.222i"

NumberFormat nf2 = NumberFormat.getInstance();
nf.setMinimumFractionDigits(1);
nf.setMaximumFractionDigits(1);

// create complex format with custom number formats
cf = new ComplexFormat(nf, nf2);
s = format.format(c); // s contains "1.110 + 2.2i"</source>
        </p>
        <p>
          Another formatting customization provided by
          <code>ComplexFormat</code> is the text used for the imaginary
          designation.  By default, the imaginary notation is "i" but, it can be
          manipulated using the <code>setImaginaryCharacter</code> method.
        </p>
        <p>
          Formatting inverse operation, parsing, can also be performed by
          <code>ComplexFormat</code>.  Parse a complex number from a string,
          simply call the <code>parse</code> method:
          <source>ComplexFormat cf = new ComplexFormat();
Complex c = cf.parse("1.110 + 2.222i");</source>
        </p>
      </subsection>
    </section>
  </body>
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