Help/Math/Complex.html

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  37 <p class="p1"><span class="s1"><b>Complex<span class="Apple-tab-span">  </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></b></span><b>complex number</b></p>
  38 <p class="p2"><br></p>
  39 <p class="p1"><b>inherits from:</b> <a href="../Core/Object.html"><span class="s2">Object</span></a> :<b> </b><a href="Magnitude.html"><span class="s3">Magnitude</span></a><span class="s3"> :</span><b> </b><a href="Number.html"><span class="s4">Number</span></a></p>
  40 <p class="p2"><br></p>
  41 <p class="p1">A class representing complex numbers.</p>
  42 <p class="p2"><br></p>
  43 <p class="p1">Note that this is a simplified representation of a complex number, which does not implement the full mathematical notion of a complex number.<span class="Apple-converted-space"> </span></p>
  44 <p class="p2"><br></p>
  45 <p class="p3"><br></p>
  46 <p class="p4"><b>Creation</b></p>
  47 <p class="p3"><br></p>
  48 <p class="p5"><b>new(real, imag)</b></p>
  49 <p class="p3"><br></p>
  50 <p class="p5"><span class="Apple-tab-span">     </span>Create a new complex number with the given real and imaginary parts.</p>
  51 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
  52 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>a = <span class="s5">Complex</span>(2, 5);</p>
  53 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>a.real;</p>
  54 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>a.imag;</p>
  55 <p class="p3"><br></p>
  56 <p class="p3"><br></p>
  57 <p class="p4"><b>Accessing</b></p>
  58 <p class="p3"><br></p>
  59 <p class="p5"><b><span class="Apple-tab-span">  </span>real</b></p>
  60 <p class="p5"><b><span class="Apple-tab-span">  </span>real_(val)</b></p>
  61 <p class="p3"><span class="Apple-tab-span">     </span></p>
  62 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>The real part of the number.</p>
  63 <p class="p3"><span class="Apple-tab-span">     </span></p>
  64 <p class="p5"><span class="Apple-tab-span">     </span><b>imag<span class="Apple-converted-space"> </span></b></p>
  65 <p class="p5"><b><span class="Apple-tab-span">  </span>imag_(val)</b></p>
  66 <p class="p3"><span class="Apple-tab-span">     </span></p>
  67 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>The imaginary part of the number.</p>
  68 <p class="p3"><span class="Apple-tab-span">     </span></p>
  69 <p class="p3"><span class="Apple-tab-span">     </span></p>
  70 <p class="p3"><br></p>
  71 <p class="p4"><b>Math</b></p>
  72 <p class="p3"><br></p>
  73 <p class="p5"><b><span class="Apple-tab-span">  </span>+ aNumber</b></p>
  74 <p class="p3"><span class="Apple-tab-span">     </span></p>
  75 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Complex addition.</p>
  76 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
  77 <p class="p7"><span class="s6"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s7">Complex</span>(2, 9) + <span class="s7">Complex</span>(-6, 2)</p>
  78 <p class="p3"><span class="Apple-tab-span">     </span></p>
  79 <p class="p5"><span class="Apple-tab-span">     </span><b>- aNumber</b></p>
  80 <p class="p3"><span class="Apple-tab-span">     </span></p>
  81 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Complex subtraction.</p>
  82 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
  83 <p class="p7"><span class="s6"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s7">Complex</span>(2, 9) - <span class="s7">Complex</span>(-6, 2)</p>
  84 <p class="p3"><span class="Apple-tab-span">     </span></p>
  85 <p class="p5"><span class="Apple-tab-span">     </span><b>* aNumber</b></p>
  86 <p class="p3"><span class="Apple-tab-span">     </span></p>
  87 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Complex multiplication.</p>
  88 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
  89 <p class="p7"><span class="s6"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s7">Complex</span>(2, 9) * <span class="s7">Complex</span>(-6, 2)</p>
  90 <p class="p3"><span class="Apple-tab-span">     </span></p>
  91 <p class="p5"><span class="Apple-tab-span">     </span><b>/ aNumber</b></p>
  92 <p class="p3"><span class="Apple-tab-span">     </span></p>
  93 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Complex division.</p>
  94 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
  95 <p class="p7"><span class="s6"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s7">Complex</span>(2, 9) / <span class="s7">Complex</span>(-6, 2)</p>
  96 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
  97 <p class="p5"><span class="s8"><span class="Apple-tab-span">    </span></span><b>** aNumber</b></p>
  98 <p class="p3"><span class="Apple-tab-span">     </span></p>
  99 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Complex exponentiation (not implemented for all combinations - some are mathematically ambiguous).</p>
 100 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 101 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="s9">Complex</span>(0, 2) ** 6</p>
 102 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>2.3 ** <span class="s5">Complex</span>(0, 2)</p>
 103 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="s10">Complex</span>(2, 9) ** 1.2 <span class="s11">// not defined</span></p>
 104 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 105 <p class="p5"><span class="s8"><span class="Apple-tab-span">    </span></span><b>exp(aNumber)</b></p>
 106 <p class="p3"><span class="Apple-tab-span">     </span></p>
 107 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Complex exponentiation with base <b>e</b>.</p>
 108 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></p>
 109 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>exp(<span class="s5">Complex</span>(2, 9))</p>
 110 <p class="p8"><span class="s12"><span class="Apple-tab-span">   </span><span class="Apple-tab-span">    </span>exp(</span><span class="s5">Complex</span><span class="s12">(0, pi)) == -1 </span>// Euler's formula: true</p>
 111 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 112 <p class="p5"><span class="Apple-tab-span">     </span><b>squared</b></p>
 113 <p class="p3"><span class="Apple-tab-span">     </span></p>
 114 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Complex self multiplication.</p>
 115 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></p>
 116 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>squared(<span class="s7">Complex</span>(2, 1))</p>
 117 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 118 <p class="p5"><span class="s8"><span class="Apple-tab-span">    </span></span><b>cubed</b></p>
 119 <p class="p3"><span class="Apple-tab-span">     </span></p>
 120 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Complex double self multiplication.</p>
 121 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></p>
 122 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>squared(<span class="s7">Complex</span>(2, 1))</p>
 123 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 124 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 125 <p class="p5"><span class="Apple-tab-span">     </span><b>&lt; aNumber</b></p>
 126 <p class="p3"><span class="Apple-tab-span">     </span></p>
 127 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer the comparison of just the real parts.</p>
 128 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 129 <p class="p7"><span class="s6"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s5">Complex</span>(2, 9) &lt; <span class="s5">Complex</span>(5, 1);</p>
 130 <p class="p3"><span class="Apple-tab-span">     </span></p>
 131 <p class="p5"><span class="Apple-tab-span">     </span><b>== aNumber</b></p>
 132 <p class="p3"><span class="Apple-tab-span">     </span></p>
 133 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer the comparison assuming that the reals (floats) are fully embedded in the complex numbers<span class="Apple-converted-space"> </span></p>
 134 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 135 <p class="p7"><span class="s6"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s5">Complex</span>(1, 0) == 1;</p>
 136 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="s5">Complex</span>(1, 5) == <span class="s5">Complex</span>(1, 5);</p>
 137 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 138 <p class="p5"><span class="Apple-tab-span">     </span><b>neg</b></p>
 139 <p class="p3"><span class="Apple-tab-span">     </span></p>
 140 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Negation of both parts.</p>
 141 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 142 <p class="p7"><span class="s6"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s7">Complex</span>(2, 9).neg</p>
 143 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 144 <p class="p5"><span class="s8"><span class="Apple-tab-span">    </span></span><b>abs</b></p>
 145 <p class="p3"><span class="Apple-tab-span">     </span></p>
 146 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>The absoulte value of a complex number is its magnitude.</p>
 147 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span></p>
 148 <p class="p7"><span class="s6"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s13">Complex</span>(3, 4).abs</p>
 149 <p class="p3"><span class="Apple-tab-span">     </span></p>
 150 <p class="p5"><span class="Apple-tab-span">     </span><b>conjugate</b></p>
 151 <p class="p3"><span class="Apple-tab-span">     </span></p>
 152 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer the complex conjugate.</p>
 153 <p class="p3"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></p>
 154 <p class="p7"><span class="s6"><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span></span><span class="s7">Complex</span>(2, 9).conjugate</p>
 155 <p class="p3"><br></p>
 156 <p class="p3"><br></p>
 157 <p class="p4"><b>Conversion</b></p>
 158 <p class="p3"><br></p>
 159 <p class="p5"><b><span class="Apple-tab-span">  </span>magnitude</b></p>
 160 <p class="p3"><span class="Apple-tab-span">     </span></p>
 161 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer the distance to the origin.</p>
 162 <p class="p3"><span class="Apple-tab-span">     </span></p>
 163 <p class="p5"><span class="Apple-tab-span">     </span><b>rho</b></p>
 164 <p class="p3"><span class="Apple-tab-span">     </span></p>
 165 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer the distance to the origin.</p>
 166 <p class="p3"><span class="Apple-tab-span">     </span></p>
 167 <p class="p5"><span class="Apple-tab-span">     </span><b>angle</b></p>
 168 <p class="p3"><span class="Apple-tab-span">     </span></p>
 169 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer the angle in radians.</p>
 170 <p class="p3"><span class="Apple-tab-span">     </span></p>
 171 <p class="p5"><span class="Apple-tab-span">     </span><b>phase</b></p>
 172 <p class="p3"><span class="Apple-tab-span">     </span></p>
 173 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer the angle in radians.</p>
 174 <p class="p3"><span class="Apple-tab-span">     </span></p>
 175 <p class="p5"><span class="Apple-tab-span">     </span><b>theta</b></p>
 176 <p class="p3"><span class="Apple-tab-span">     </span></p>
 177 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer the angle in radians.</p>
 178 <p class="p3"><span class="Apple-tab-span">     </span></p>
 179 <p class="p5"><span class="Apple-tab-span">     </span><b>asPoint</b></p>
 180 <p class="p3"><span class="Apple-tab-span">     </span></p>
 181 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Convert to a <a href="../Geometry/Point.html"><span class="s3">Point</span></a>.</p>
 182 <p class="p3"><span class="Apple-tab-span">     </span></p>
 183 <p class="p5"><span class="Apple-tab-span">     </span><b>asPolar</b></p>
 184 <p class="p3"><span class="Apple-tab-span">     </span></p>
 185 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Convert to a <a href="Polar.html"><span class="s3">Polar</span></a></p>
 186 <p class="p3"><span class="Apple-tab-span">     </span></p>
 187 <p class="p5"><span class="Apple-tab-span">     </span><b>asInteger</b></p>
 188 <p class="p3"><span class="Apple-tab-span">     </span></p>
 189 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer real part as <a href="Integer.html"><span class="s3">Integer</span></a>.</p>
 190 <p class="p3"><span class="Apple-tab-span">     </span></p>
 191 <p class="p5"><span class="Apple-tab-span">     </span><b>asFloat</b></p>
 192 <p class="p3"><span class="Apple-tab-span">     </span></p>
 193 <p class="p5"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>Answer real part as <a href="Float.html"><span class="s3">Float</span></a>.</p>
 194 <p class="p3"><span class="Apple-tab-span">     </span></p>
 195 <p class="p3"><br></p>
 196 <p class="p3"><br></p>
 197 <p class="p3"><br></p>
 198 <p class="p9">// example</p>
 199 <p class="p6"><br></p>
 200 <p class="p7">a = <span class="s4">Complex</span>(0, 1);</p>
 201 <p class="p9"><span class="s12">a * a; </span>// returns Complex(-1, 0);</p>
 202 <p class="p6"><br></p>
 203 <p class="p9">// julia set approximation</p>
 204 <p class="p7">f = { <span class="s4">|z|</span> z * z + <span class="s4">Complex</span>(0.70176, 0.3842) };</p>
 205 <p class="p6"><br></p>
 206 <p class="p7">(</p>
 207 <p class="p7"><span class="Apple-tab-span">     </span><span class="s4">var</span> n = 80, xs = 400, ys = 400, dx = xs / n, dy = ys / n, zoom = 3, offset = -0.5;</p>
 208 <p class="p7"><span class="Apple-tab-span">     </span><span class="s4">var</span> field = { <span class="s4">|x|</span> { <span class="s4">|y|</span> <span class="s4">Complex</span>(x / n + offset * zoom, y / n + offset * zoom) } ! n } ! n;</p>
 209 <p class="p7"><span class="Apple-tab-span">     </span>w = <span class="s4">Window</span>(<span class="s14">"Julia set"</span>, bounds:<span class="s4">Rect</span>(200, 200, xs, ys)).front;</p>
 210 <p class="p7"><span class="Apple-tab-span">     </span>w.view.background_(<span class="s4">Color</span>.black);</p>
 211 <p class="p6"><span class="Apple-tab-span">     </span></p>
 212 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>w.drawHook = {</p>
 213 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span>n.do { <span class="s4">|x|</span></p>
 214 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span>n.do { <span class="s4">|y|</span></p>
 215 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="s4">var</span> z = field[x][y];</p>
 216 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span>z = f.(z);</p>
 217 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span>field[x][y] = z;</p>
 218 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="s4">Pen</span>.color = <span class="s4">Color</span>.gray(z.rho.linlin(-100, 100, 1, 0));</p>
 219 <p class="p6"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-converted-space"> </span></p>
 220 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="s4">Pen</span>.addRect(</p>
 221 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="s4">Rect</span>(x * dx, y * dy, dx, dy)</p>
 222 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span>);</p>
 223 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="s4">Pen</span>.fill</p>
 224 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span>}</p>
 225 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span><span class="Apple-tab-span">    </span>}</p>
 226 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>};</p>
 227 <p class="p7"><span class="Apple-tab-span">     </span><span class="Apple-tab-span">    </span>fork({ 6.do { w.refresh; 2.wait } }, <span class="s4">AppClock</span>)</p>
 228 <p class="p7">)</p>
 229 <p class="p6"><br></p>
 230 <p class="p6"><br></p>
 231 <p class="p6"><br></p>
 232 <p class="p3"><br></p>
 233 </body>
 234 </html>