1Math::Complex(3pm)     Perl Programmers Reference Guide     Math::Complex(3pm)
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NAME

6       Math::Complex - complex numbers and associated mathematical functions
7

SYNOPSIS

9               use Math::Complex;
10
11               $z = Math::Complex->make(5, 6);
12               $t = 4 - 3*i + $z;
13               $j = cplxe(1, 2*pi/3);
14

DESCRIPTION

16       This package lets you create and manipulate complex numbers. By
17       default, Perl limits itself to real numbers, but an extra "use"
18       statement brings full complex support, along with a full set of
19       mathematical functions typically associated with and/or extended to
20       complex numbers.
21
22       If you wonder what complex numbers are, they were invented to be able
23       to solve the following equation:
24
25               x*x = -1
26
27       and by definition, the solution is noted i (engineers use j instead
28       since i usually denotes an intensity, but the name does not matter).
29       The number i is a pure imaginary number.
30
31       The arithmetics with pure imaginary numbers works just like you would
32       expect it with real numbers... you just have to remember that
33
34               i*i = -1
35
36       so you have:
37
38               5i + 7i = i * (5 + 7) = 12i
39               4i - 3i = i * (4 - 3) = i
40               4i * 2i = -8
41               6i / 2i = 3
42               1 / i = -i
43
44       Complex numbers are numbers that have both a real part and an imaginary
45       part, and are usually noted:
46
47               a + bi
48
49       where "a" is the real part and "b" is the imaginary part. The
50       arithmetic with complex numbers is straightforward. You have to keep
51       track of the real and the imaginary parts, but otherwise the rules used
52       for real numbers just apply:
53
54               (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
55               (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
56
57       A graphical representation of complex numbers is possible in a plane
58       (also called the complex plane, but it's really a 2D plane).  The
59       number
60
61               z = a + bi
62
63       is the point whose coordinates are (a, b). Actually, it would be the
64       vector originating from (0, 0) to (a, b). It follows that the addition
65       of two complex numbers is a vectorial addition.
66
67       Since there is a bijection between a point in the 2D plane and a
68       complex number (i.e. the mapping is unique and reciprocal), a complex
69       number can also be uniquely identified with polar coordinates:
70
71               [rho, theta]
72
73       where "rho" is the distance to the origin, and "theta" the angle
74       between the vector and the x axis. There is a notation for this using
75       the exponential form, which is:
76
77               rho * exp(i * theta)
78
79       where i is the famous imaginary number introduced above. Conversion
80       between this form and the cartesian form "a + bi" is immediate:
81
82               a = rho * cos(theta)
83               b = rho * sin(theta)
84
85       which is also expressed by this formula:
86
87               z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
88
89       In other words, it's the projection of the vector onto the x and y
90       axes. Mathematicians call rho the norm or modulus and theta the
91       argument of the complex number. The norm of "z" is marked here as
92       abs(z).
93
94       The polar notation (also known as the trigonometric representation) is
95       much more handy for performing multiplications and divisions of complex
96       numbers, whilst the cartesian notation is better suited for additions
97       and subtractions. Real numbers are on the x axis, and therefore y or
98       theta is zero or pi.
99
100       All the common operations that can be performed on a real number have
101       been defined to work on complex numbers as well, and are merely
102       extensions of the operations defined on real numbers. This means they
103       keep their natural meaning when there is no imaginary part, provided
104       the number is within their definition set.
105
106       For instance, the "sqrt" routine which computes the square root of its
107       argument is only defined for non-negative real numbers and yields a
108       non-negative real number (it is an application from R+ to R+).  If we
109       allow it to return a complex number, then it can be extended to
110       negative real numbers to become an application from R to C (the set of
111       complex numbers):
112
113               sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
114
115       It can also be extended to be an application from C to C, whilst its
116       restriction to R behaves as defined above by using the following
117       definition:
118
119               sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
120
121       Indeed, a negative real number can be noted "[x,pi]" (the modulus x is
122       always non-negative, so "[x,pi]" is really "-x", a negative number) and
123       the above definition states that
124
125               sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
126
127       which is exactly what we had defined for negative real numbers above.
128       The "sqrt" returns only one of the solutions: if you want the both, use
129       the "root" function.
130
131       All the common mathematical functions defined on real numbers that are
132       extended to complex numbers share that same property of working as
133       usual when the imaginary part is zero (otherwise, it would not be
134       called an extension, would it?).
135
136       A new operation possible on a complex number that is the identity for
137       real numbers is called the conjugate, and is noted with a horizontal
138       bar above the number, or "~z" here.
139
140                z = a + bi
141               ~z = a - bi
142
143       Simple... Now look:
144
145               z * ~z = (a + bi) * (a - bi) = a*a + b*b
146
147       We saw that the norm of "z" was noted abs(z) and was defined as the
148       distance to the origin, also known as:
149
150               rho = abs(z) = sqrt(a*a + b*b)
151
152       so
153
154               z * ~z = abs(z) ** 2
155
156       If z is a pure real number (i.e. "b == 0"), then the above yields:
157
158               a * a = abs(a) ** 2
159
160       which is true ("abs" has the regular meaning for real number, i.e.
161       stands for the absolute value). This example explains why the norm of
162       "z" is noted abs(z): it extends the "abs" function to complex numbers,
163       yet is the regular "abs" we know when the complex number actually has
164       no imaginary part... This justifies a posteriori our use of the "abs"
165       notation for the norm.
166

OPERATIONS

168       Given the following notations:
169
170               z1 = a + bi = r1 * exp(i * t1)
171               z2 = c + di = r2 * exp(i * t2)
172               z = <any complex or real number>
173
174       the following (overloaded) operations are supported on complex numbers:
175
176               z1 + z2 = (a + c) + i(b + d)
177               z1 - z2 = (a - c) + i(b - d)
178               z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
179               z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
180               z1 ** z2 = exp(z2 * log z1)
181               ~z = a - bi
182               abs(z) = r1 = sqrt(a*a + b*b)
183               sqrt(z) = sqrt(r1) * exp(i * t/2)
184               exp(z) = exp(a) * exp(i * b)
185               log(z) = log(r1) + i*t
186               sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
187               cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
188               atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.
189
190       The definition used for complex arguments of atan2() is
191
192              -i log((x + iy)/sqrt(x*x+y*y))
193
194       Note that atan2(0, 0) is not well-defined.
195
196       The following extra operations are supported on both real and complex
197       numbers:
198
199               Re(z) = a
200               Im(z) = b
201               arg(z) = t
202               abs(z) = r
203
204               cbrt(z) = z ** (1/3)
205               log10(z) = log(z) / log(10)
206               logn(z, n) = log(z) / log(n)
207
208               tan(z) = sin(z) / cos(z)
209
210               csc(z) = 1 / sin(z)
211               sec(z) = 1 / cos(z)
212               cot(z) = 1 / tan(z)
213
214               asin(z) = -i * log(i*z + sqrt(1-z*z))
215               acos(z) = -i * log(z + i*sqrt(1-z*z))
216               atan(z) = i/2 * log((i+z) / (i-z))
217
218               acsc(z) = asin(1 / z)
219               asec(z) = acos(1 / z)
220               acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))
221
222               sinh(z) = 1/2 (exp(z) - exp(-z))
223               cosh(z) = 1/2 (exp(z) + exp(-z))
224               tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))
225
226               csch(z) = 1 / sinh(z)
227               sech(z) = 1 / cosh(z)
228               coth(z) = 1 / tanh(z)
229
230               asinh(z) = log(z + sqrt(z*z+1))
231               acosh(z) = log(z + sqrt(z*z-1))
232               atanh(z) = 1/2 * log((1+z) / (1-z))
233
234               acsch(z) = asinh(1 / z)
235               asech(z) = acosh(1 / z)
236               acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
237
238       arg, abs, log, csc, cot, acsc, acot, csch, coth, acosech, acotanh, have
239       aliases rho, theta, ln, cosec, cotan, acosec, acotan, cosech, cotanh,
240       acosech, acotanh, respectively.  "Re", "Im", "arg", "abs", "rho", and
241       "theta" can be used also as mutators.  The "cbrt" returns only one of
242       the solutions: if you want all three, use the "root" function.
243
244       The root function is available to compute all the n roots of some
245       complex, where n is a strictly positive integer.  There are exactly n
246       such roots, returned as a list. Getting the number mathematicians call
247       "j" such that:
248
249               1 + j + j*j = 0;
250
251       is a simple matter of writing:
252
253               $j = (root(1, 3))[1];
254
255       The kth root for "z = [r,t]" is given by:
256
257               (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
258
259       You can return the kth root directly by "root(z, n, k)", indexing
260       starting from zero and ending at n - 1.
261
262       The spaceship numeric comparison operator, <=>, is also defined. In
263       order to ensure its restriction to real numbers is conform to what you
264       would expect, the comparison is run on the real part of the complex
265       number first, and imaginary parts are compared only when the real parts
266       match.
267

CREATION

269       To create a complex number, use either:
270
271               $z = Math::Complex->make(3, 4);
272               $z = cplx(3, 4);
273
274       if you know the cartesian form of the number, or
275
276               $z = 3 + 4*i;
277
278       if you like. To create a number using the polar form, use either:
279
280               $z = Math::Complex->emake(5, pi/3);
281               $x = cplxe(5, pi/3);
282
283       instead. The first argument is the modulus, the second is the angle (in
284       radians, the full circle is 2*pi).  (Mnemonic: "e" is used as a
285       notation for complex numbers in the polar form).
286
287       It is possible to write:
288
289               $x = cplxe(-3, pi/4);
290
291       but that will be silently converted into "[3,-3pi/4]", since the
292       modulus must be non-negative (it represents the distance to the origin
293       in the complex plane).
294
295       It is also possible to have a complex number as either argument of the
296       "make", "emake", "cplx", and "cplxe": the appropriate component of the
297       argument will be used.
298
299               $z1 = cplx(-2,  1);
300               $z2 = cplx($z1, 4);
301
302       The "new", "make", "emake", "cplx", and "cplxe" will also understand a
303       single (string) argument of the forms
304
305               2-3i
306               -3i
307               [2,3]
308               [2,-3pi/4]
309               [2]
310
311       in which case the appropriate cartesian and exponential components will
312       be parsed from the string and used to create new complex numbers.  The
313       imaginary component and the theta, respectively, will default to zero.
314
315       The "new", "make", "emake", "cplx", and "cplxe" will also understand
316       the case of no arguments: this means plain zero or (0, 0).
317

DISPLAYING

319       When printed, a complex number is usually shown under its cartesian
320       style a+bi, but there are legitimate cases where the polar style [r,t]
321       is more appropriate.  The process of converting the complex number into
322       a string that can be displayed is known as stringification.
323
324       By calling the class method "Math::Complex::display_format" and
325       supplying either "polar" or "cartesian" as an argument, you override
326       the default display style, which is "cartesian". Not supplying any
327       argument returns the current settings.
328
329       This default can be overridden on a per-number basis by calling the
330       "display_format" method instead. As before, not supplying any argument
331       returns the current display style for this number. Otherwise whatever
332       you specify will be the new display style for this particular number.
333
334       For instance:
335
336               use Math::Complex;
337
338               Math::Complex::display_format('polar');
339               $j = (root(1, 3))[1];
340               print "j = $j\n";               # Prints "j = [1,2pi/3]"
341               $j->display_format('cartesian');
342               print "j = $j\n";               # Prints "j = -0.5+0.866025403784439i"
343
344       The polar style attempts to emphasize arguments like k*pi/n (where n is
345       a positive integer and k an integer within [-9, +9]), this is called
346       polar pretty-printing.
347
348       For the reverse of stringifying, see the "make" and "emake".
349
350   CHANGED IN PERL 5.6
351       The "display_format" class method and the corresponding
352       "display_format" object method can now be called using a parameter hash
353       instead of just a one parameter.
354
355       The old display format style, which can have values "cartesian" or
356       "polar", can be changed using the "style" parameter.
357
358               $j->display_format(style => "polar");
359
360       The one parameter calling convention also still works.
361
362               $j->display_format("polar");
363
364       There are two new display parameters.
365
366       The first one is "format", which is a sprintf()-style format string to
367       be used for both numeric parts of the complex number(s).  The is
368       somewhat system-dependent but most often it corresponds to "%.15g".
369       You can revert to the default by setting the "format" to "undef".
370
371               # the $j from the above example
372
373               $j->display_format('format' => '%.5f');
374               print "j = $j\n";               # Prints "j = -0.50000+0.86603i"
375               $j->display_format('format' => undef);
376               print "j = $j\n";               # Prints "j = -0.5+0.86603i"
377
378       Notice that this affects also the return values of the "display_format"
379       methods: in list context the whole parameter hash will be returned, as
380       opposed to only the style parameter value.  This is a potential
381       incompatibility with earlier versions if you have been calling the
382       "display_format" method in list context.
383
384       The second new display parameter is "polar_pretty_print", which can be
385       set to true or false, the default being true.  See the previous section
386       for what this means.
387

USAGE

389       Thanks to overloading, the handling of arithmetics with complex numbers
390       is simple and almost transparent.
391
392       Here are some examples:
393
394               use Math::Complex;
395
396               $j = cplxe(1, 2*pi/3);  # $j ** 3 == 1
397               print "j = $j, j**3 = ", $j ** 3, "\n";
398               print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
399
400               $z = -16 + 0*i;                 # Force it to be a complex
401               print "sqrt($z) = ", sqrt($z), "\n";
402
403               $k = exp(i * 2*pi/3);
404               print "$j - $k = ", $j - $k, "\n";
405
406               $z->Re(3);                      # Re, Im, arg, abs,
407               $j->arg(2);                     # (the last two aka rho, theta)
408                                               # can be used also as mutators.
409

CONSTANTS

411   PI
412       The constant "pi" and some handy multiples of it (pi2, pi4, and pip2
413       (pi/2) and pip4 (pi/4)) are also available if separately exported:
414
415           use Math::Complex ':pi';
416           $third_of_circle = pi2 / 3;
417
418   Inf
419       The floating point infinity can be exported as a subroutine Inf():
420
421           use Math::Complex qw(Inf sinh);
422           my $AlsoInf = Inf() + 42;
423           my $AnotherInf = sinh(1e42);
424           print "$AlsoInf is $AnotherInf\n" if $AlsoInf == $AnotherInf;
425
426       Note that the stringified form of infinity varies between platforms: it
427       can be for example any of
428
429          inf
430          infinity
431          INF
432          1.#INF
433
434       or it can be something else.
435
436       Also note that in some platforms trying to use the infinity in
437       arithmetic operations may result in Perl crashing because using an
438       infinity causes SIGFPE or its moral equivalent to be sent.  The way to
439       ignore this is
440
441         local $SIG{FPE} = sub { };
442

ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO

444       The division (/) and the following functions
445
446               log     ln      log10   logn
447               tan     sec     csc     cot
448               atan    asec    acsc    acot
449               tanh    sech    csch    coth
450               atanh   asech   acsch   acoth
451
452       cannot be computed for all arguments because that would mean dividing
453       by zero or taking logarithm of zero. These situations cause fatal
454       runtime errors looking like this
455
456               cot(0): Division by zero.
457               (Because in the definition of cot(0), the divisor sin(0) is 0)
458               Died at ...
459
460       or
461
462               atanh(-1): Logarithm of zero.
463               Died at...
464
465       For the "csc", "cot", "asec", "acsc", "acot", "csch", "coth", "asech",
466       "acsch", the argument cannot be 0 (zero).  For the logarithmic
467       functions and the "atanh", "acoth", the argument cannot be 1 (one).
468       For the "atanh", "acoth", the argument cannot be -1 (minus one).  For
469       the "atan", "acot", the argument cannot be "i" (the imaginary unit).
470       For the "atan", "acoth", the argument cannot be "-i" (the negative
471       imaginary unit).  For the "tan", "sec", "tanh", the argument cannot be
472       pi/2 + k * pi, where k is any integer.  atan2(0, 0) is undefined, and
473       if the complex arguments are used for atan2(), a division by zero will
474       happen if z1**2+z2**2 == 0.
475
476       Note that because we are operating on approximations of real numbers,
477       these errors can happen when merely `too close' to the singularities
478       listed above.
479

ERRORS DUE TO INDIGESTIBLE ARGUMENTS

481       The "make" and "emake" accept both real and complex arguments.  When
482       they cannot recognize the arguments they will die with error messages
483       like the following
484
485           Math::Complex::make: Cannot take real part of ...
486           Math::Complex::make: Cannot take real part of ...
487           Math::Complex::emake: Cannot take rho of ...
488           Math::Complex::emake: Cannot take theta of ...
489

BUGS

491       Saying "use Math::Complex;" exports many mathematical routines in the
492       caller environment and even overrides some ("sqrt", "log", "atan2").
493       This is construed as a feature by the Authors, actually... ;-)
494
495       All routines expect to be given real or complex numbers. Don't attempt
496       to use BigFloat, since Perl has currently no rule to disambiguate a '+'
497       operation (for instance) between two overloaded entities.
498
499       In Cray UNICOS there is some strange numerical instability that results
500       in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast.  Beware.
501       The bug may be in UNICOS math libs, in UNICOS C compiler, in
502       Math::Complex.  Whatever it is, it does not manifest itself anywhere
503       else where Perl runs.
504

SEE ALSO

506       Math::Trig
507

AUTHORS

509       Daniel S. Lewart <lewart!at!uiuc.edu>, Jarkko Hietaniemi
510       <jhi!at!iki.fi>, Raphael Manfredi <Raphael_Manfredi!at!pobox.com>,
511       Zefram <zefram@fysh.org>
512

LICENSE

514       This library is free software; you can redistribute it and/or modify it
515       under the same terms as Perl itself.
516
517
518
519perl v5.38.2                      2023-11-30                Math::Complex(3pm)
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