mingw/html/lib/Math/Complex.html

   1 <?xml version="1.0" ?>
   2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
   3 <html xmlns="http://www.w3.org/1999/xhtml">
   4 <head>
   5 <title>Math::Complex - complex numbers and associated mathematical functions</title>
   6 <meta http-equiv="content-type" content="text/html; charset=utf-8" />
   7 <link rev="made" href="mailto:" />
   8 </head>
   9
  10 <body style="background-color: white">
  11 <table border="0" width="100%" cellspacing="0" cellpadding="3">
  12 <tr><td class="block" style="background-color: #cccccc" valign="middle">
  13 <big><strong><span class="block">&nbsp;Math::Complex - complex numbers and associated mathematical functions</span></strong></big>
  14 </td></tr>
  15 </table>
  16
  17 <p><a name="__index__"></a></p>
  18 <!-- INDEX BEGIN -->
  19
  20 <ul>
  21
  22         <li><a href="#name">NAME</a></li>
  23         <li><a href="#synopsis">SYNOPSIS</a></li>
  24         <li><a href="#description">DESCRIPTION</a></li>
  25         <li><a href="#operations">OPERATIONS</a></li>
  26         <li><a href="#creation">CREATION</a></li>
  27         <li><a href="#displaying">DISPLAYING</a></li>
  28         <ul>
  29
  30                 <li><a href="#changed_in_perl_5_6">CHANGED IN PERL 5.6</a></li>
  31         </ul>
  32
  33         <li><a href="#usage">USAGE</a></li>
  34         <li><a href="#errors_due_to_division_by_zero_or_logarithm_of_zero">ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO</a></li>
  35         <li><a href="#errors_due_to_indigestible_arguments">ERRORS DUE TO INDIGESTIBLE ARGUMENTS</a></li>
  36         <li><a href="#bugs">BUGS</a></li>
  37         <li><a href="#authors">AUTHORS</a></li>
  38 </ul>
  39 <!-- INDEX END -->
  40
  41 <hr />
  42 <p>
  43 </p>
  44 <hr />
  45 <h1><a name="name">NAME</a></h1>
  46 <p>Math::Complex - complex numbers and associated mathematical functions</p>
  47 <p>
  48 </p>
  49 <hr />
  50 <h1><a name="synopsis">SYNOPSIS</a></h1>
  51 <pre>
  52         use Math::Complex;</pre>
  53 <pre>
  54         $z = Math::Complex-&gt;make(5, 6);
  55         $t = 4 - 3*i + $z;
  56         $j = cplxe(1, 2*pi/3);</pre>
  57 <p>
  58 </p>
  59 <hr />
  60 <h1><a name="description">DESCRIPTION</a></h1>
  61 <p>This package lets you create and manipulate complex numbers. By default,
  62 <em>Perl</em> limits itself to real numbers, but an extra <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_use"><code>use</code></a> statement brings
  63 full complex support, along with a full set of mathematical functions
  64 typically associated with and/or extended to complex numbers.</p>
  65 <p>If you wonder what complex numbers are, they were invented to be able to solve
  66 the following equation:</p>
  67 <pre>
  68         x*x = -1</pre>
  69 <p>and by definition, the solution is noted <em>i</em> (engineers use <em>j</em> instead since
  70 <em>i</em> usually denotes an intensity, but the name does not matter). The number
  71 <em>i</em> is a pure <em>imaginary</em> number.</p>
  72 <p>The arithmetics with pure imaginary numbers works just like you would expect
  73 it with real numbers... you just have to remember that</p>
  74 <pre>
  75         i*i = -1</pre>
  76 <p>so you have:</p>
  77 <pre>
  78         5i + 7i = i * (5 + 7) = 12i
  79         4i - 3i = i * (4 - 3) = i
  80         4i * 2i = -8
  81         6i / 2i = 3
  82         1 / i = -i</pre>
  83 <p>Complex numbers are numbers that have both a real part and an imaginary
  84 part, and are usually noted:</p>
  85 <pre>
  86         a + bi</pre>
  87 <p>where <code>a</code> is the <em>real</em> part and <a href="file://C|\msysgit\mingw\html/pod/perlguts.html#item_b"><code>b</code></a> is the <em>imaginary</em> part. The
  88 arithmetic with complex numbers is straightforward. You have to
  89 keep track of the real and the imaginary parts, but otherwise the
  90 rules used for real numbers just apply:</p>
  91 <pre>
  92         (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
  93         (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i</pre>
  94 <p>A graphical representation of complex numbers is possible in a plane
  95 (also called the <em>complex plane</em>, but it's really a 2D plane).
  96 The number</p>
  97 <pre>
  98         z = a + bi</pre>
  99 <p>is the point whose coordinates are (a, b). Actually, it would
 100 be the vector originating from (0, 0) to (a, b). It follows that the addition
 101 of two complex numbers is a vectorial addition.</p>
 102 <p>Since there is a bijection between a point in the 2D plane and a complex
 103 number (i.e. the mapping is unique and reciprocal), a complex number
 104 can also be uniquely identified with polar coordinates:</p>
 105 <pre>
 106         [rho, theta]</pre>
 107 <p>where <code>rho</code> is the distance to the origin, and <code>theta</code> the angle between
 108 the vector and the <em>x</em> axis. There is a notation for this using the
 109 exponential form, which is:</p>
 110 <pre>
 111         rho * exp(i * theta)</pre>
 112 <p>where <em>i</em> is the famous imaginary number introduced above. Conversion
 113 between this form and the cartesian form <code>a + bi</code> is immediate:</p>
 114 <pre>
 115         a = rho * cos(theta)
 116         b = rho * sin(theta)</pre>
 117 <p>which is also expressed by this formula:</p>
 118 <pre>
 119         z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)</pre>
 120 <p>In other words, it's the projection of the vector onto the <em>x</em> and <em>y</em>
 121 axes. Mathematicians call <em>rho</em> the <em>norm</em> or <em>modulus</em> and <em>theta</em>
 122 the <em>argument</em> of the complex number. The <em>norm</em> of <code>z</code> will be
 123 noted <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_abs"><code>abs(z)</code></a>.</p>
 124 <p>The polar notation (also known as the trigonometric
 125 representation) is much more handy for performing multiplications and
 126 divisions of complex numbers, whilst the cartesian notation is better
 127 suited for additions and subtractions. Real numbers are on the <em>x</em>
 128 axis, and therefore <em>theta</em> is zero or <em>pi</em>.</p>
 129 <p>All the common operations that can be performed on a real number have
 130 been defined to work on complex numbers as well, and are merely
 131 <em>extensions</em> of the operations defined on real numbers. This means
 132 they keep their natural meaning when there is no imaginary part, provided
 133 the number is within their definition set.</p>
 134 <p>For instance, the <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_sqrt"><code>sqrt</code></a> routine which computes the square root of
 135 its argument is only defined for non-negative real numbers and yields a
 136 non-negative real number (it is an application from <strong>R+</strong> to <strong>R+</strong>).
 137 If we allow it to return a complex number, then it can be extended to
 138 negative real numbers to become an application from <strong>R</strong> to <strong>C</strong> (the
 139 set of complex numbers):</p>
 140 <pre>
 141         sqrt(x) = x &gt;= 0 ? sqrt(x) : sqrt(-x)*i</pre>
 142 <p>It can also be extended to be an application from <strong>C</strong> to <strong>C</strong>,
 143 whilst its restriction to <strong>R</strong> behaves as defined above by using
 144 the following definition:</p>
 145 <pre>
 146         sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)</pre>
 147 <p>Indeed, a negative real number can be noted <code>[x,pi]</code> (the modulus
 148 <em>x</em> is always non-negative, so <code>[x,pi]</code> is really <code>-x</code>, a negative
 149 number) and the above definition states that</p>
 150 <pre>
 151         sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i</pre>
 152 <p>which is exactly what we had defined for negative real numbers above.
 153 The <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_sqrt"><code>sqrt</code></a> returns only one of the solutions: if you want the both,
 154 use the <code>root</code> function.</p>
 155 <p>All the common mathematical functions defined on real numbers that
 156 are extended to complex numbers share that same property of working
 157 <em>as usual</em> when the imaginary part is zero (otherwise, it would not
 158 be called an extension, would it?).</p>
 159 <p>A <em>new</em> operation possible on a complex number that is
 160 the identity for real numbers is called the <em>conjugate</em>, and is noted
 161 with a horizontal bar above the number, or <code>~z</code> here.</p>
 162 <pre>
 163          z = a + bi
 164         ~z = a - bi</pre>
 165 <p>Simple... Now look:</p>
 166 <pre>
 167         z * ~z = (a + bi) * (a - bi) = a*a + b*b</pre>
 168 <p>We saw that the norm of <code>z</code> was noted <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_abs"><code>abs(z)</code></a> and was defined as the
 169 distance to the origin, also known as:</p>
 170 <pre>
 171         rho = abs(z) = sqrt(a*a + b*b)</pre>
 172 <p>so</p>
 173 <pre>
 174         z * ~z = abs(z) ** 2</pre>
 175 <p>If z is a pure real number (i.e. <code>b == 0</code>), then the above yields:</p>
 176 <pre>
 177         a * a = abs(a) ** 2</pre>
 178 <p>which is true (<a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_abs"><code>abs</code></a> has the regular meaning for real number, i.e. stands
 179 for the absolute value). This example explains why the norm of <code>z</code> is
 180 noted <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_abs"><code>abs(z)</code></a>: it extends the <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_abs"><code>abs</code></a> function to complex numbers, yet
 181 is the regular <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_abs"><code>abs</code></a> we know when the complex number actually has no
 182 imaginary part... This justifies <em>a posteriori</em> our use of the <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_abs"><code>abs</code></a>
 183 notation for the norm.</p>
 184 <p>
 185 </p>
 186 <hr />
 187 <h1><a name="operations">OPERATIONS</a></h1>
 188 <p>Given the following notations:</p>
 189 <pre>
 190         z1 = a + bi = r1 * exp(i * t1)
 191         z2 = c + di = r2 * exp(i * t2)
 192         z = &lt;any complex or real number&gt;</pre>
 193 <p>the following (overloaded) operations are supported on complex numbers:</p>
 194 <pre>
 195         z1 + z2 = (a + c) + i(b + d)
 196         z1 - z2 = (a - c) + i(b - d)
 197         z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
 198         z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
 199         z1 ** z2 = exp(z2 * log z1)
 200         ~z = a - bi
 201         abs(z) = r1 = sqrt(a*a + b*b)
 202         sqrt(z) = sqrt(r1) * exp(i * t/2)
 203         exp(z) = exp(a) * exp(i * b)
 204         log(z) = log(r1) + i*t
 205         sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
 206         cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
 207         atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.</pre>
 208 <p>The definition used for complex arguments of <code>atan2()</code> is</p>
 209 <pre>
 210        -i log((x + iy)/sqrt(x*x+y*y))</pre>
 211 <p>The following extra operations are supported on both real and complex
 212 numbers:</p>
 213 <pre>
 214         Re(z) = a
 215         Im(z) = b
 216         arg(z) = t
 217         abs(z) = r</pre>
 218 <pre>
 219         cbrt(z) = z ** (1/3)
 220         log10(z) = log(z) / log(10)
 221         logn(z, n) = log(z) / log(n)</pre>
 222 <pre>
 223         tan(z) = sin(z) / cos(z)</pre>
 224 <pre>
 225         csc(z) = 1 / sin(z)
 226         sec(z) = 1 / cos(z)
 227         cot(z) = 1 / tan(z)</pre>
 228 <pre>
 229         asin(z) = -i * log(i*z + sqrt(1-z*z))
 230         acos(z) = -i * log(z + i*sqrt(1-z*z))
 231         atan(z) = i/2 * log((i+z) / (i-z))</pre>
 232 <pre>
 233         acsc(z) = asin(1 / z)
 234         asec(z) = acos(1 / z)
 235         acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))</pre>
 236 <pre>
 237         sinh(z) = 1/2 (exp(z) - exp(-z))
 238         cosh(z) = 1/2 (exp(z) + exp(-z))
 239         tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))</pre>
 240 <pre>
 241         csch(z) = 1 / sinh(z)
 242         sech(z) = 1 / cosh(z)
 243         coth(z) = 1 / tanh(z)</pre>
 244 <pre>
 245         asinh(z) = log(z + sqrt(z*z+1))
 246         acosh(z) = log(z + sqrt(z*z-1))
 247         atanh(z) = 1/2 * log((1+z) / (1-z))</pre>
 248 <pre>
 249         acsch(z) = asinh(1 / z)
 250         asech(z) = acosh(1 / z)
 251         acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))</pre>
 252 <p><em>arg</em>, <em>abs</em>, <em>log</em>, <em>csc</em>, <em>cot</em>, <em>acsc</em>, <em>acot</em>, <em>csch</em>,
 253 <em>coth</em>, <em>acosech</em>, <em>acotanh</em>, have aliases <em>rho</em>, <em>theta</em>, <em>ln</em>,
 254 <em>cosec</em>, <em>cotan</em>, <em>acosec</em>, <em>acotan</em>, <em>cosech</em>, <em>cotanh</em>,
 255 <em>acosech</em>, <em>acotanh</em>, respectively.  <code>Re</code>, <code>Im</code>, <code>arg</code>, <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_abs"><code>abs</code></a>,
 256 <code>rho</code>, and <code>theta</code> can be used also as mutators.  The <code>cbrt</code>
 257 returns only one of the solutions: if you want all three, use the
 258 <code>root</code> function.</p>
 259 <p>The <em>root</em> function is available to compute all the <em>n</em>
 260 roots of some complex, where <em>n</em> is a strictly positive integer.
 261 There are exactly <em>n</em> such roots, returned as a list. Getting the
 262 number mathematicians call <code>j</code> such that:</p>
 263 <pre>
 264         1 + j + j*j = 0;</pre>
 265 <p>is a simple matter of writing:</p>
 266 <pre>
 267         $j = ((root(1, 3))[1];</pre>
 268 <p>The <em>k</em>th root for <code>z = [r,t]</code> is given by:</p>
 269 <pre>
 270         (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)</pre>
 271 <p>You can return the <em>k</em>th root directly by <code>root(z, n, k)</code>,
 272 indexing starting from <em>zero</em> and ending at <em>n - 1</em>.</p>
 273 <p>The <em>spaceship</em> comparison operator, &lt;=&gt;, is also defined. In
 274 order to ensure its restriction to real numbers is conform to what you
 275 would expect, the comparison is run on the real part of the complex
 276 number first, and imaginary parts are compared only when the real
 277 parts match.</p>
 278 <p>
 279 </p>
 280 <hr />
 281 <h1><a name="creation">CREATION</a></h1>
 282 <p>To create a complex number, use either:</p>
 283 <pre>
 284         $z = Math::Complex-&gt;make(3, 4);
 285         $z = cplx(3, 4);</pre>
 286 <p>if you know the cartesian form of the number, or</p>
 287 <pre>
 288         $z = 3 + 4*i;</pre>
 289 <p>if you like. To create a number using the polar form, use either:</p>
 290 <pre>
 291         $z = Math::Complex-&gt;emake(5, pi/3);
 292         $x = cplxe(5, pi/3);</pre>
 293 <p>instead. The first argument is the modulus, the second is the angle
 294 (in radians, the full circle is 2*pi).  (Mnemonic: <code>e</code> is used as a
 295 notation for complex numbers in the polar form).</p>
 296 <p>It is possible to write:</p>
 297 <pre>
 298         $x = cplxe(-3, pi/4);</pre>
 299 <p>but that will be silently converted into <code>[3,-3pi/4]</code>, since the
 300 modulus must be non-negative (it represents the distance to the origin
 301 in the complex plane).</p>
 302 <p>It is also possible to have a complex number as either argument of the
 303 <code>make</code>, <code>emake</code>, <code>cplx</code>, and <code>cplxe</code>: the appropriate component of
 304 the argument will be used.</p>
 305 <pre>
 306         $z1 = cplx(-2,  1);
 307         $z2 = cplx($z1, 4);</pre>
 308 <p>The <code>new</code>, <code>make</code>, <code>emake</code>, <code>cplx</code>, and <code>cplxe</code> will also
 309 understand a single (string) argument of the forms</p>
 310 <pre>
 311         2-3i
 312         -3i
 313         [2,3]
 314         [2,-3pi/4]
 315         [2]</pre>
 316 <p>in which case the appropriate cartesian and exponential components
 317 will be parsed from the string and used to create new complex numbers.
 318 The imaginary component and the theta, respectively, will default to zero.</p>
 319 <p>The <code>new</code>, <code>make</code>, <code>emake</code>, <code>cplx</code>, and <code>cplxe</code> will also
 320 understand the case of no arguments: this means plain zero or (0, 0).</p>
 321 <p>
 322 </p>
 323 <hr />
 324 <h1><a name="displaying">DISPLAYING</a></h1>
 325 <p>When printed, a complex number is usually shown under its cartesian
 326 style <em>a+bi</em>, but there are legitimate cases where the polar style
 327 <em>[r,t]</em> is more appropriate.  The process of converting the complex
 328 number into a string that can be displayed is known as <em>stringification</em>.</p>
 329 <p>By calling the class method <code>Math::Complex::display_format</code> and
 330 supplying either <code>&quot;polar&quot;</code> or <code>&quot;cartesian&quot;</code> as an argument, you
 331 override the default display style, which is <code>&quot;cartesian&quot;</code>. Not
 332 supplying any argument returns the current settings.</p>
 333 <p>This default can be overridden on a per-number basis by calling the
 334 <code>display_format</code> method instead. As before, not supplying any argument
 335 returns the current display style for this number. Otherwise whatever you
 336 specify will be the new display style for <em>this</em> particular number.</p>
 337 <p>For instance:</p>
 338 <pre>
 339         use Math::Complex;</pre>
 340 <pre>
 341         Math::Complex::display_format('polar');
 342         $j = (root(1, 3))[1];
 343         print &quot;j = $j\n&quot;;               # Prints &quot;j = [1,2pi/3]&quot;
 344         $j-&gt;display_format('cartesian');
 345         print &quot;j = $j\n&quot;;               # Prints &quot;j = -0.5+0.866025403784439i&quot;</pre>
 346 <p>The polar style attempts to emphasize arguments like <em>k*pi/n</em>
 347 (where <em>n</em> is a positive integer and <em>k</em> an integer within [-9, +9]),
 348 this is called <em>polar pretty-printing</em>.</p>
 349 <p>For the reverse of stringifying, see the <code>make</code> and <code>emake</code>.</p>
 350 <p>
 351 </p>
 352 <h2><a name="changed_in_perl_5_6">CHANGED IN PERL 5.6</a></h2>
 353 <p>The <code>display_format</code> class method and the corresponding
 354 <code>display_format</code> object method can now be called using
 355 a parameter hash instead of just a one parameter.</p>
 356 <p>The old display format style, which can have values <code>&quot;cartesian&quot;</code> or
 357 <code>&quot;polar&quot;</code>, can be changed using the <code>&quot;style&quot;</code> parameter.</p>
 358 <pre>
 359         $j-&gt;display_format(style =&gt; &quot;polar&quot;);</pre>
 360 <p>The one parameter calling convention also still works.</p>
 361 <pre>
 362         $j-&gt;display_format(&quot;polar&quot;);</pre>
 363 <p>There are two new display parameters.</p>
 364 <p>The first one is <code>&quot;format&quot;</code>, which is a sprintf()-style format string
 365 to be used for both numeric parts of the complex number(s).  The is
 366 somewhat system-dependent but most often it corresponds to <code>&quot;%.15g&quot;</code>.
 367 You can revert to the default by setting the <code>format</code> to <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_undef"><code>undef</code></a>.</p>
 368 <pre>
 369         # the $j from the above example</pre>
 370 <pre>
 371         $j-&gt;display_format('format' =&gt; '%.5f');
 372         print &quot;j = $j\n&quot;;               # Prints &quot;j = -0.50000+0.86603i&quot;
 373         $j-&gt;display_format('format' =&gt; undef);
 374         print &quot;j = $j\n&quot;;               # Prints &quot;j = -0.5+0.86603i&quot;</pre>
 375 <p>Notice that this affects also the return values of the
 376 <code>display_format</code> methods: in list context the whole parameter hash
 377 will be returned, as opposed to only the style parameter value.
 378 This is a potential incompatibility with earlier versions if you
 379 have been calling the <code>display_format</code> method in list context.</p>
 380 <p>The second new display parameter is <code>&quot;polar_pretty_print&quot;</code>, which can
 381 be set to true or false, the default being true.  See the previous
 382 section for what this means.</p>
 383 <p>
 384 </p>
 385 <hr />
 386 <h1><a name="usage">USAGE</a></h1>
 387 <p>Thanks to overloading, the handling of arithmetics with complex numbers
 388 is simple and almost transparent.</p>
 389 <p>Here are some examples:</p>
 390 <pre>
 391         use Math::Complex;</pre>
 392 <pre>
 393         $j = cplxe(1, 2*pi/3);  # $j ** 3 == 1
 394         print &quot;j = $j, j**3 = &quot;, $j ** 3, &quot;\n&quot;;
 395         print &quot;1 + j + j**2 = &quot;, 1 + $j + $j**2, &quot;\n&quot;;</pre>
 396 <pre>
 397         $z = -16 + 0*i;                 # Force it to be a complex
 398         print &quot;sqrt($z) = &quot;, sqrt($z), &quot;\n&quot;;</pre>
 399 <pre>
 400         $k = exp(i * 2*pi/3);
 401         print &quot;$j - $k = &quot;, $j - $k, &quot;\n&quot;;</pre>
 402 <pre>
 403         $z-&gt;Re(3);                      # Re, Im, arg, abs,
 404         $j-&gt;arg(2);                     # (the last two aka rho, theta)
 405                                         # can be used also as mutators.</pre>
 406 <p>
 407 </p>
 408 <hr />
 409 <h1><a name="errors_due_to_division_by_zero_or_logarithm_of_zero">ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO</a></h1>
 410 <p>The division (/) and the following functions</p>
 411 <pre>
 412         log     ln      log10   logn
 413         tan     sec     csc     cot
 414         atan    asec    acsc    acot
 415         tanh    sech    csch    coth
 416         atanh   asech   acsch   acoth</pre>
 417 <p>cannot be computed for all arguments because that would mean dividing
 418 by zero or taking logarithm of zero. These situations cause fatal
 419 runtime errors looking like this</p>
 420 <pre>
 421         cot(0): Division by zero.
 422         (Because in the definition of cot(0), the divisor sin(0) is 0)
 423         Died at ...</pre>
 424 <p>or</p>
 425 <pre>
 426         atanh(-1): Logarithm of zero.
 427         Died at...</pre>
 428 <p>For the <code>csc</code>, <code>cot</code>, <code>asec</code>, <code>acsc</code>, <code>acot</code>, <code>csch</code>, <code>coth</code>,
 429 <code>asech</code>, <code>acsch</code>, the argument cannot be <code>0</code> (zero).  For the
 430 logarithmic functions and the <code>atanh</code>, <code>acoth</code>, the argument cannot
 431 be <code>1</code> (one).  For the <code>atanh</code>, <code>acoth</code>, the argument cannot be
 432 <code>-1</code> (minus one).  For the <code>atan</code>, <code>acot</code>, the argument cannot be
 433 <code>i</code> (the imaginary unit).  For the <code>atan</code>, <code>acoth</code>, the argument
 434 cannot be <code>-i</code> (the negative imaginary unit).  For the <code>tan</code>,
 435 <code>sec</code>, <code>tanh</code>, the argument cannot be <em>pi/2 + k * pi</em>, where <em>k</em>
 436 is any integer.  atan2(0, 0) is undefined, and if the complex arguments
 437 are used for atan2(), a division by zero will happen if z1**2+z2**2 == 0.</p>
 438 <p>Note that because we are operating on approximations of real numbers,
 439 these errors can happen when merely `too close' to the singularities
 440 listed above.</p>
 441 <p>
 442 </p>
 443 <hr />
 444 <h1><a name="errors_due_to_indigestible_arguments">ERRORS DUE TO INDIGESTIBLE ARGUMENTS</a></h1>
 445 <p>The <code>make</code> and <code>emake</code> accept both real and complex arguments.
 446 When they cannot recognize the arguments they will die with error
 447 messages like the following</p>
 448 <pre>
 449     Math::Complex::make: Cannot take real part of ...
 450     Math::Complex::make: Cannot take real part of ...
 451     Math::Complex::emake: Cannot take rho of ...
 452     Math::Complex::emake: Cannot take theta of ...</pre>
 453 <p>
 454 </p>
 455 <hr />
 456 <h1><a name="bugs">BUGS</a></h1>
 457 <p>Saying <code>use Math::Complex;</code> exports many mathematical routines in the
 458 caller environment and even overrides some (<a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_sqrt"><code>sqrt</code></a>, <a href="file://C|\msysgit\mingw\html/pod/perlfunc.html#item_log"><code>log</code></a>, <code>atan2</code>).
 459 This is construed as a feature by the Authors, actually... ;-)</p>
 460 <p>All routines expect to be given real or complex numbers. Don't attempt to
 461 use BigFloat, since Perl has currently no rule to disambiguate a '+'
 462 operation (for instance) between two overloaded entities.</p>
 463 <p>In Cray UNICOS there is some strange numerical instability that results
 464 in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast.  Beware.
 465 The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex.
 466 Whatever it is, it does not manifest itself anywhere else where Perl runs.</p>
 467 <p>
 468 </p>
 469 <hr />
 470 <h1><a name="authors">AUTHORS</a></h1>
 471 <p>Daniel S. Lewart &lt;<em><a href="mailto:d-lewart@uiuc.edu">d-lewart@uiuc.edu</a></em>&gt;</p>
 472 <p>Original authors Raphael Manfredi &lt;<em><a href="mailto:Raphael_Manfredi@pobox.com">Raphael_Manfredi@pobox.com</a></em>&gt; and
 473 Jarkko Hietaniemi &lt;<em><a href="mailto:jhi@iki.fi">jhi@iki.fi</a></em>&gt;</p>
 474 <table border="0" width="100%" cellspacing="0" cellpadding="3">
 475 <tr><td class="block" style="background-color: #cccccc" valign="middle">
 476 <big><strong><span class="block">&nbsp;Math::Complex - complex numbers and associated mathematical functions</span></strong></big>
 477 </td></tr>
 478 </table>
 479
 480 </body>
 481
 482 </html>