1Math::Complex(3pm) Perl Programmers Reference Guide Math::Complex(3pm)
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6 Math::Complex - complex numbers and associated mathematical functions
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9 use Math::Complex;
10 $z = Math::Complex->make(5, 6);
12 $t = 4 - 3*i + $z;
13 $j = cplxe(1, 2*pi/3);
14
16 This package lets you create and manipulate complex numbers. By
17 default, Perl limits itself to real numbers, but an extra "use" state‐
18 ment brings full complex support, along with a full set of mathematical
19 functions typically associated with and/or extended to complex numbers.
20 If you wonder what complex numbers are, they were invented to be able
22 to solve the following equation:
23 x*x = -1
25 and by definition, the solution is noted i (engineers use j instead
27 since i usually denotes an intensity, but the name does not matter).
28 The number i is a pure imaginary number.
29 The arithmetics with pure imaginary numbers works just like you would
31 expect it with real numbers... you just have to remember that
32 i*i = -1
34 so you have:
36 5i + 7i = i * (5 + 7) = 12i
38 4i - 3i = i * (4 - 3) = i
39 4i * 2i = -8
40 6i / 2i = 3
41 1 / i = -i
42 Complex numbers are numbers that have both a real part and an imaginary
44 part, and are usually noted:
45 a + bi
47 where "a" is the real part and "b" is the imaginary part. The arith‐
49 metic with complex numbers is straightforward. You have to keep track
50 of the real and the imaginary parts, but otherwise the rules used for
51 real numbers just apply:
52 (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
54 (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
55 A graphical representation of complex numbers is possible in a plane
57 (also called the complex plane, but it's really a 2D plane). The num‐
58 ber
59 z = a + bi
61 is the point whose coordinates are (a, b). Actually, it would be the
63 vector originating from (0, 0) to (a, b). It follows that the addition
64 of two complex numbers is a vectorial addition.
65 Since there is a bijection between a point in the 2D plane and a com‐
67 plex number (i.e. the mapping is unique and reciprocal), a complex num‐
68 ber can also be uniquely identified with polar coordinates:
69 [rho, theta]
71 where "rho" is the distance to the origin, and "theta" the angle
73 between the vector and the x axis. There is a notation for this using
74 the exponential form, which is:
75 rho * exp(i * theta)
77 where i is the famous imaginary number introduced above. Conversion
79 between this form and the cartesian form "a + bi" is immediate:
80 a = rho * cos(theta)
82 b = rho * sin(theta)
83 which is also expressed by this formula:
85 z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
87 In other words, it's the projection of the vector onto the x and y
89 axes. Mathematicians call rho the norm or modulus and theta the argu‐
90 ment of the complex number. The norm of "z" will be noted abs(z).
91 The polar notation (also known as the trigonometric representation) is
93 much more handy for performing multiplications and divisions of complex
94 numbers, whilst the cartesian notation is better suited for additions
95 and subtractions. Real numbers are on the x axis, and therefore theta
96 is zero or pi.
97 All the common operations that can be performed on a real number have
99 been defined to work on complex numbers as well, and are merely exten‐
100 sions of the operations defined on real numbers. This means they keep
101 their natural meaning when there is no imaginary part, provided the
102 number is within their definition set.
103 For instance, the "sqrt" routine which computes the square root of its
105 argument is only defined for non-negative real numbers and yields a
106 non-negative real number (it is an application from R+ to R+). If we
107 allow it to return a complex number, then it can be extended to nega‐
108 tive real numbers to become an application from R to C (the set of com‐
109 plex numbers):
110 sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
112 It can also be extended to be an application from C to C, whilst its
114 restriction to R behaves as defined above by using the following defi‐
115 nition:
116 sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
118 Indeed, a negative real number can be noted "[x,pi]" (the modulus x is
120 always non-negative, so "[x,pi]" is really "-x", a negative number) and
121 the above definition states that
122 sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
124 which is exactly what we had defined for negative real numbers above.
126 The "sqrt" returns only one of the solutions: if you want the both, use
127 the "root" function.
128 All the common mathematical functions defined on real numbers that are
130 extended to complex numbers share that same property of working as
131 usual when the imaginary part is zero (otherwise, it would not be
132 called an extension, would it?).
133 A new operation possible on a complex number that is the identity for
135 real numbers is called the conjugate, and is noted with a horizontal
136 bar above the number, or "~z" here.
137 z = a + bi
139 ~z = a - bi
140 Simple... Now look:
142 z * ~z = (a + bi) * (a - bi) = a*a + b*b
144 We saw that the norm of "z" was noted abs(z) and was defined as the
146 distance to the origin, also known as:
147 rho = abs(z) = sqrt(a*a + b*b)
149 so
151 z * ~z = abs(z) ** 2
153 If z is a pure real number (i.e. "b == 0"), then the above yields:
155 a * a = abs(a) ** 2
157 which is true ("abs" has the regular meaning for real number, i.e.
159 stands for the absolute value). This example explains why the norm of
160 "z" is noted abs(z): it extends the "abs" function to complex numbers,
161 yet is the regular "abs" we know when the complex number actually has
162 no imaginary part... This justifies a posteriori our use of the "abs"
163 notation for the norm.
164
166 Given the following notations:
167 z1 = a + bi = r1 * exp(i * t1)
169 z2 = c + di = r2 * exp(i * t2)
170 z = <any complex or real number>
171 the following (overloaded) operations are supported on complex numbers:
173 z1 + z2 = (a + c) + i(b + d)
175 z1 - z2 = (a - c) + i(b - d)
176 z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
177 z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
178 z1 ** z2 = exp(z2 * log z1)
179 ~z = a - bi
180 abs(z) = r1 = sqrt(a*a + b*b)
181 sqrt(z) = sqrt(r1) * exp(i * t/2)
182 exp(z) = exp(a) * exp(i * b)
183 log(z) = log(r1) + i*t
184 sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
185 cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
186 atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.
187 The definition used for complex arguments of atan2() is
189 -i log((x + iy)/sqrt(x*x+y*y))
191 The following extra operations are supported on both real and complex
193 numbers:
194 Re(z) = a
196 Im(z) = b
197 arg(z) = t
198 abs(z) = r
199 cbrt(z) = z ** (1/3)
201 log10(z) = log(z) / log(10)
202 logn(z, n) = log(z) / log(n)
203 tan(z) = sin(z) / cos(z)
205 csc(z) = 1 / sin(z)
207 sec(z) = 1 / cos(z)
208 cot(z) = 1 / tan(z)
209 asin(z) = -i * log(i*z + sqrt(1-z*z))
211 acos(z) = -i * log(z + i*sqrt(1-z*z))
212 atan(z) = i/2 * log((i+z) / (i-z))
213 acsc(z) = asin(1 / z)
215 asec(z) = acos(1 / z)
216 acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))
217 sinh(z) = 1/2 (exp(z) - exp(-z))
219 cosh(z) = 1/2 (exp(z) + exp(-z))
220 tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))
221 csch(z) = 1 / sinh(z)
223 sech(z) = 1 / cosh(z)
224 coth(z) = 1 / tanh(z)
225 asinh(z) = log(z + sqrt(z*z+1))
227 acosh(z) = log(z + sqrt(z*z-1))
228 atanh(z) = 1/2 * log((1+z) / (1-z))
229 acsch(z) = asinh(1 / z)
231 asech(z) = acosh(1 / z)
232 acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
233 arg, abs, log, csc, cot, acsc, acot, csch, coth, acosech, acotanh, have
235 aliases rho, theta, ln, cosec, cotan, acosec, acotan, cosech, cotanh,
236 acosech, acotanh, respectively. "Re", "Im", "arg", "abs", "rho", and
237 "theta" can be used also as mutators. The "cbrt" returns only one of
238 the solutions: if you want all three, use the "root" function.
239 The root function is available to compute all the n roots of some com‐
241 plex, where n is a strictly positive integer. There are exactly n such
242 roots, returned as a list. Getting the number mathematicians call "j"
243 such that:
244 1 + j + j*j = 0;
246 is a simple matter of writing:
248 $j = ((root(1, 3))[1];
250 The kth root for "z = [r,t]" is given by:
252 (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
254 You can return the kth root directly by "root(z, n, k)", indexing
256 starting from zero and ending at n - 1.
257 The spaceship comparison operator, <=>, is also defined. In order to
259 ensure its restriction to real numbers is conform to what you would
260 expect, the comparison is run on the real part of the complex number
261 first, and imaginary parts are compared only when the real parts match.
262
264 To create a complex number, use either:
265 $z = Math::Complex->make(3, 4);
267 $z = cplx(3, 4);
268 if you know the cartesian form of the number, or
270 $z = 3 + 4*i;
272 if you like. To create a number using the polar form, use either:
274 $z = Math::Complex->emake(5, pi/3);
276 $x = cplxe(5, pi/3);
277 instead. The first argument is the modulus, the second is the angle (in
279 radians, the full circle is 2*pi). (Mnemonic: "e" is used as a nota‐
280 tion for complex numbers in the polar form).
281 It is possible to write:
283 $x = cplxe(-3, pi/4);
285 but that will be silently converted into "[3,-3pi/4]", since the modu‐
287 lus must be non-negative (it represents the distance to the origin in
288 the complex plane).
289 It is also possible to have a complex number as either argument of the
291 "make", "emake", "cplx", and "cplxe": the appropriate component of the
292 argument will be used.
293 $z1 = cplx(-2, 1);
295 $z2 = cplx($z1, 4);
296 The "new", "make", "emake", "cplx", and "cplxe" will also understand a
298 single (string) argument of the forms
299 2-3i
301 -3i
302 [2,3]
303 [2,-3pi/4]
304 [2]
305 in which case the appropriate cartesian and exponential components will
307 be parsed from the string and used to create new complex numbers. The
308 imaginary component and the theta, respectively, will default to zero.
309 The "new", "make", "emake", "cplx", and "cplxe" will also understand
311 the case of no arguments: this means plain zero or (0, 0).
312
314 When printed, a complex number is usually shown under its cartesian
315 style a+bi, but there are legitimate cases where the polar style [r,t]
316 is more appropriate. The process of converting the complex number into
317 a string that can be displayed is known as stringification.
318 By calling the class method "Math::Complex::display_format" and supply‐
320 ing either "polar" or "cartesian" as an argument, you override the
321 default display style, which is "cartesian". Not supplying any argument
322 returns the current settings.
323 This default can be overridden on a per-number basis by calling the
325 "display_format" method instead. As before, not supplying any argument
326 returns the current display style for this number. Otherwise whatever
327 you specify will be the new display style for this particular number.
328 For instance:
330 use Math::Complex;
332 Math::Complex::display_format('polar');
334 $j = (root(1, 3))[1];
335 print "j = $j\n"; # Prints "j = [1,2pi/3]"
336 $j->display_format('cartesian');
337 print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
338 The polar style attempts to emphasize arguments like k*pi/n (where n is
340 a positive integer and k an integer within [-9, +9]), this is called
341 polar pretty-printing.
342 For the reverse of stringifying, see the "make" and "emake".
344 CHANGED IN PERL 5.6
346 The "display_format" class method and the corresponding "display_for‐
348 mat" object method can now be called using a parameter hash instead of
349 just a one parameter.
350 The old display format style, which can have values "cartesian" or
352 "polar", can be changed using the "style" parameter.
353 $j->display_format(style => "polar");
355 The one parameter calling convention also still works.
357 $j->display_format("polar");
359 There are two new display parameters.
361 The first one is "format", which is a sprintf()-style format string to
363 be used for both numeric parts of the complex number(s). The is some‐
364 what system-dependent but most often it corresponds to "%.15g". You
365 can revert to the default by setting the "format" to "undef".
366 # the $j from the above example
368 $j->display_format('format' => '%.5f');
370 print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
371 $j->display_format('format' => undef);
372 print "j = $j\n"; # Prints "j = -0.5+0.86603i"
373 Notice that this affects also the return values of the "display_format"
375 methods: in list context the whole parameter hash will be returned, as
376 opposed to only the style parameter value. This is a potential incom‐
377 patibility with earlier versions if you have been calling the "dis‐
378 play_format" method in list context.
379 The second new display parameter is "polar_pretty_print", which can be
381 set to true or false, the default being true. See the previous section
382 for what this means.
383
385 Thanks to overloading, the handling of arithmetics with complex numbers
386 is simple and almost transparent.
387 Here are some examples:
389 use Math::Complex;
391 $j = cplxe(1, 2*pi/3); # $j ** 3 == 1
393 print "j = $j, j**3 = ", $j ** 3, "\n";
394 print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
395 $z = -16 + 0*i; # Force it to be a complex
397 print "sqrt($z) = ", sqrt($z), "\n";
398 $k = exp(i * 2*pi/3);
400 print "$j - $k = ", $j - $k, "\n";
401 $z->Re(3); # Re, Im, arg, abs,
403 $j->arg(2); # (the last two aka rho, theta)
404 # can be used also as mutators.
405
407 The division (/) and the following functions
408 log ln log10 logn
410 tan sec csc cot
411 atan asec acsc acot
412 tanh sech csch coth
413 atanh asech acsch acoth
414 cannot be computed for all arguments because that would mean dividing
416 by zero or taking logarithm of zero. These situations cause fatal run‐
417 time errors looking like this
418 cot(0): Division by zero.
420 (Because in the definition of cot(0), the divisor sin(0) is 0)
421 Died at ...
422 or
424 atanh(-1): Logarithm of zero.
426 Died at...
427 For the "csc", "cot", "asec", "acsc", "acot", "csch", "coth", "asech",
429 "acsch", the argument cannot be 0 (zero). For the logarithmic func‐
430 tions and the "atanh", "acoth", the argument cannot be 1 (one). For
431 the "atanh", "acoth", the argument cannot be "-1" (minus one). For the
432 "atan", "acot", the argument cannot be "i" (the imaginary unit). For
433 the "atan", "acoth", the argument cannot be "-i" (the negative imagi‐
434 nary unit). For the "tan", "sec", "tanh", the argument cannot be pi/2
435 + k * pi, where k is any integer. atan2(0, 0) is undefined, and if the
436 complex arguments are used for atan2(), a division by zero will happen
437 if z1**2+z2**2 == 0.
438 Note that because we are operating on approximations of real numbers,
440 these errors can happen when merely `too close' to the singularities
441 listed above.
442
444 The "make" and "emake" accept both real and complex arguments. When
445 they cannot recognize the arguments they will die with error messages
446 like the following
447 Math::Complex::make: Cannot take real part of ...
449 Math::Complex::make: Cannot take real part of ...
450 Math::Complex::emake: Cannot take rho of ...
451 Math::Complex::emake: Cannot take theta of ...
452
454 Saying "use Math::Complex;" exports many mathematical routines in the
455 caller environment and even overrides some ("sqrt", "log", "atan2").
456 This is construed as a feature by the Authors, actually... ;-)
457 All routines expect to be given real or complex numbers. Don't attempt
459 to use BigFloat, since Perl has currently no rule to disambiguate a '+'
460 operation (for instance) between two overloaded entities.
461 In Cray UNICOS there is some strange numerical instability that results
463 in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware.
464 The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Com‐
465 plex. Whatever it is, it does not manifest itself anywhere else where
466 Perl runs.
467
469 Daniel S. Lewart <d-lewart@uiuc.edu>
470 Original authors Raphael Manfredi <Raphael_Manfredi@pobox.com> and
472 Jarkko Hietaniemi <jhi@iki.fi>
473perl v5.8.8 2001-09-21 Math::Complex(3pm)