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



NAME
       Math::Complex - complex numbers and associated mathematical functions

SYNOPSIS
               use Math::Complex;

               $z = Math::Complex->make(5, 6);
               $t = 4 - 3*i + $z;
               $j = cplxe(1, 2*pi/3);

DESCRIPTION
       This package lets you create and manipulate complex numbers. By default, Perl limits itself to real
       numbers, but an extra "use" statement brings full complex support, along with a full set of
       mathematical functions typically associated with and/or extended to complex numbers.

       If you wonder what complex numbers are, they were invented to be able to solve the following
       equation:

               x*x = -1

       and by definition, the solution is noted i (engineers use j instead since i usually denotes an
       intensity, but the name does not matter). The number i is a pure imaginary number.

       The arithmetics with pure imaginary numbers works just like you would expect it with real numbers...
       you just have to remember that

               i*i = -1

       so you have:

               5i + 7i = i * (5 + 7) = 12i
               4i - 3i = i * (4 - 3) = i
               4i * 2i = -8
               6i / 2i = 3
               1 / i = -i

       Complex numbers are numbers that have both a real part and an imaginary part, and are usually noted:

               a + bi

       where "a" is the real part and "b" is the imaginary part. The arithmetic with complex numbers is
       straightforward. You have to keep track of the real and the imaginary parts, but otherwise the rules
       used for real numbers just apply:

               (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
               (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i

       A graphical representation of complex numbers is possible in a plane (also called the complex plane,
       but it's really a 2D plane).  The number

               z = a + bi

       is the point whose coordinates are (a, b). Actually, it would be the vector originating from (0, 0)
       to (a, b). It follows that the addition of two complex numbers is a vectorial addition.

       Since there is a bijection between a point in the 2D plane and a complex number (i.e. the mapping is
       unique and reciprocal), a complex number can also be uniquely identified with polar coordinates:

               [rho, theta]

       where "rho" is the distance to the origin, and "theta" the angle between the vector and the x axis.
       There is a notation for this using the exponential form, which is:

               rho * exp(i * theta)

       where i is the famous imaginary number introduced above. Conversion between this form and the
       cartesian form "a + bi" is immediate:

               a = rho * cos(theta)
               b = rho * sin(theta)

       which is also expressed by this formula:

               z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)

       In other words, it's the projection of the vector onto the x and y axes. Mathematicians call rho the
       norm or modulus and theta the argument of the complex number. The norm of "z" is marked here as
       abs(z).

       The polar notation (also known as the trigonometric representation) is much more handy for performing
       multiplications and divisions of complex numbers, whilst the cartesian notation is better suited for
       additions and subtractions. Real numbers are on the x axis, and therefore y or theta is zero or pi.

       All the common operations that can be performed on a real number have been defined to work on complex
       numbers as well, and are merely extensions of the operations defined on real numbers. This means they
       keep their natural meaning when there is no imaginary part, provided the number is within their
       definition set.

       For instance, the "sqrt" routine which computes the square root of its argument is only defined for
       non-negative real numbers and yields a non-negative real number (it is an application from R+ to R+).
       If we allow it to return a complex number, then it can be extended to negative real numbers to become
       an application from R to C (the set of complex numbers):

               sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i

       It can also be extended to be an application from C to C, whilst its restriction to R behaves as
       defined above by using the following definition:

               sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)

       Indeed, a negative real number can be noted "[x,pi]" (the modulus x is always non-negative, so
       "[x,pi]" is really "-x", a negative number) and the above definition states that

               sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i

       which is exactly what we had defined for negative real numbers above.  The "sqrt" returns only one of
       the solutions: if you want the both, use the "root" function.

       All the common mathematical functions defined on real numbers that are extended to complex numbers
       share that same property of working as usual when the imaginary part is zero (otherwise, it would not
       be called an extension, would it?).

       A new operation possible on a complex number that is the identity for real numbers is called the
       conjugate, and is noted with a horizontal bar above the number, or "~z" here.

                z = a + bi
               ~z = a - bi

       Simple... Now look:

               z * ~z = (a + bi) * (a - bi) = a*a + b*b

       We saw that the norm of "z" was noted abs(z) and was defined as the distance to the origin, also
       known as:

               rho = abs(z) = sqrt(a*a + b*b)

       so

               z * ~z = abs(z) ** 2

       If z is a pure real number (i.e. "b == 0"), then the above yields:

               a * a = abs(a) ** 2

       which is true ("abs" has the regular meaning for real number, i.e. stands for the absolute value).
       This example explains why the norm of "z" is noted abs(z): it extends the "abs" function to complex
       numbers, yet is the regular "abs" we know when the complex number actually has no imaginary part...
       This justifies a posteriori our use of the "abs" notation for the norm.

OPERATIONS
       Given the following notations:

               z1 = a + bi = r1 * exp(i * t1)
               z2 = c + di = r2 * exp(i * t2)
               z = <any complex or real number>

       the following (overloaded) operations are supported on complex numbers:

               z1 + z2 = (a + c) + i(b + d)
               z1 - z2 = (a - c) + i(b - d)
               z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
               z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
               z1 ** z2 = exp(z2 * log z1)
               ~z = a - bi
               abs(z) = r1 = sqrt(a*a + b*b)
               sqrt(z) = sqrt(r1) * exp(i * t/2)
               exp(z) = exp(a) * exp(i * b)
               log(z) = log(r1) + i*t
               sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
               cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
               atan2(y, x) = atan(y / x) # Minding the right quadrant, note the order.

       The definition used for complex arguments of atan2() is

              -i log((x + iy)/sqrt(x*x+y*y))

       Note that atan2(0, 0) is not well-defined.

       The following extra operations are supported on both real and complex numbers:

               Re(z) = a
               Im(z) = b
               arg(z) = t
               abs(z) = r

               cbrt(z) = z ** (1/3)
               log10(z) = log(z) / log(10)
               logn(z, n) = log(z) / log(n)

               tan(z) = sin(z) / cos(z)

               csc(z) = 1 / sin(z)
               sec(z) = 1 / cos(z)
               cot(z) = 1 / tan(z)

               asin(z) = -i * log(i*z + sqrt(1-z*z))
               acos(z) = -i * log(z + i*sqrt(1-z*z))
               atan(z) = i/2 * log((i+z) / (i-z))

               acsc(z) = asin(1 / z)
               asec(z) = acos(1 / z)
               acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))

               sinh(z) = 1/2 (exp(z) - exp(-z))
               cosh(z) = 1/2 (exp(z) + exp(-z))
               tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))

               csch(z) = 1 / sinh(z)
               sech(z) = 1 / cosh(z)
               coth(z) = 1 / tanh(z)

               asinh(z) = log(z + sqrt(z*z+1))
               acosh(z) = log(z + sqrt(z*z-1))
               atanh(z) = 1/2 * log((1+z) / (1-z))

               acsch(z) = asinh(1 / z)
               asech(z) = acosh(1 / z)
               acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))

       arg, abs, log, csc, cot, acsc, acot, csch, coth, acosech, acotanh, have aliases rho, theta, ln,
       cosec, cotan, acosec, acotan, cosech, cotanh, acosech, acotanh, respectively.  "Re", "Im", "arg",
       "abs", "rho", and "theta" can be used also as mutators.  The "cbrt" returns only one of the
       solutions: if you want all three, use the "root" function.

       The root function is available to compute all the n roots of some complex, where n is a strictly
       positive integer.  There are exactly n such roots, returned as a list. Getting the number
       mathematicians call "j" such that:

               1 + j + j*j = 0;

       is a simple matter of writing:

               $j = ((root(1, 3))[1];

       The kth root for "z = [r,t]" is given by:

               (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)

       You can return the kth root directly by "root(z, n, k)", indexing starting from zero and ending at n
       - 1.

       The spaceship numeric comparison operator, <=>, is also defined. In order to ensure its restriction
       to real numbers is conform to what you would expect, the comparison is run on the real part of the
       complex number first, and imaginary parts are compared only when the real parts match.

CREATION
       To create a complex number, use either:

               $z = Math::Complex->make(3, 4);
               $z = cplx(3, 4);

       if you know the cartesian form of the number, or

               $z = 3 + 4*i;

       if you like. To create a number using the polar form, use either:

               $z = Math::Complex->emake(5, pi/3);
               $x = cplxe(5, pi/3);

       instead. The first argument is the modulus, the second is the angle (in radians, the full circle is
       2*pi).  (Mnemonic: "e" is used as a notation for complex numbers in the polar form).

       It is possible to write:

               $x = cplxe(-3, pi/4);

       but that will be silently converted into "[3,-3pi/4]", since the modulus must be non-negative (it
       represents the distance to the origin in the complex plane).

       It is also possible to have a complex number as either argument of the "make", "emake", "cplx", and
       "cplxe": the appropriate component of the argument will be used.

               $z1 = cplx(-2,  1);
               $z2 = cplx($z1, 4);

       The "new", "make", "emake", "cplx", and "cplxe" will also understand a single (string) argument of
       the forms

               2-3i
               -3i
               [2,3]
               [2,-3pi/4]
               [2]

       in which case the appropriate cartesian and exponential components will be parsed from the string and
       used to create new complex numbers.  The imaginary component and the theta, respectively, will
       default to zero.

       The "new", "make", "emake", "cplx", and "cplxe" will also understand the case of no arguments: this
       means plain zero or (0, 0).

DISPLAYING
       When printed, a complex number is usually shown under its cartesian style a+bi, but there are
       legitimate cases where the polar style [r,t] is more appropriate.  The process of converting the
       complex number into a string that can be displayed is known as stringification.

       By calling the class method "Math::Complex::display_format" and supplying either "polar" or
       "cartesian" as an argument, you override the default display style, which is "cartesian". Not
       supplying any argument returns the current settings.

       This default can be overridden on a per-number basis by calling the "display_format" method instead.
       As before, not supplying any argument returns the current display style for this number. Otherwise
       whatever you specify will be the new display style for this particular number.

       For instance:

               use Math::Complex;

               Math::Complex::display_format('polar');
               $j = (root(1, 3))[1];
               print "j = $j\n";               # Prints "j = [1,2pi/3]"
               $j->display_format('cartesian');
               print "j = $j\n";               # Prints "j = -0.5+0.866025403784439i"

       The polar style attempts to emphasize arguments like k*pi/n (where n is a positive integer and k an
       integer within [-9, +9]), this is called polar pretty-printing.

       For the reverse of stringifying, see the "make" and "emake".

   CHANGED IN PERL 5.6
       The "display_format" class method and the corresponding "display_format" object method can now be
       called using a parameter hash instead of just a one parameter.

       The old display format style, which can have values "cartesian" or "polar", can be changed using the
       "style" parameter.

               $j->display_format(style => "polar");

       The one parameter calling convention also still works.

               $j->display_format("polar");

       There are two new display parameters.

       The first one is "format", which is a sprintf()-style format string to be used for both numeric parts
       of the complex number(s).  The is somewhat system-dependent but most often it corresponds to "%.15g".
       You can revert to the default by setting the "format" to "undef".

               # the $j from the above example

               $j->display_format('format' => '%.5f');
               print "j = $j\n";               # Prints "j = -0.50000+0.86603i"
               $j->display_format('format' => undef);
               print "j = $j\n";               # Prints "j = -0.5+0.86603i"

       Notice that this affects also the return values of the "display_format" methods: in list context the
       whole parameter hash will be returned, as opposed to only the style parameter value.  This is a
       potential incompatibility with earlier versions if you have been calling the "display_format" method
       in list context.

       The second new display parameter is "polar_pretty_print", which can be set to true or false, the
       default being true.  See the previous section for what this means.

USAGE
       Thanks to overloading, the handling of arithmetics with complex numbers is simple and almost
       transparent.

       Here are some examples:

               use Math::Complex;

               $j = cplxe(1, 2*pi/3);  # $j ** 3 == 1
               print "j = $j, j**3 = ", $j ** 3, "\n";
               print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";

               $z = -16 + 0*i;                 # Force it to be a complex
               print "sqrt($z) = ", sqrt($z), "\n";

               $k = exp(i * 2*pi/3);
               print "$j - $k = ", $j - $k, "\n";

               $z->Re(3);                      # Re, Im, arg, abs,
               $j->arg(2);                     # (the last two aka rho, theta)
                                               # can be used also as mutators.

CONSTANTS
   PI
       The constant "pi" and some handy multiples of it (pi2, pi4, and pip2 (pi/2) and pip4 (pi/4)) are also
       available if separately exported:

           use Math::Complex ':pi';
           $third_of_circle = pi2 / 3;

   Inf
       The floating point infinity can be exported as a subroutine Inf():

           use Math::Complex qw(Inf sinh);
           my $AlsoInf = Inf() + 42;
           my $AnotherInf = sinh(1e42);
           print "$AlsoInf is $AnotherInf\n" if $AlsoInf == $AnotherInf;

       Note that the stringified form of infinity varies between platforms: it can be for example any of

          inf
          infinity
          INF
          1.#INF

       or it can be something else.

       Also note that in some platforms trying to use the infinity in arithmetic operations may result in
       Perl crashing because using an infinity causes SIGFPE or its moral equivalent to be sent.  The way to
       ignore this is

         local $SIG{FPE} = sub { };

ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
       The division (/) and the following functions

               log     ln      log10   logn
               tan     sec     csc     cot
               atan    asec    acsc    acot
               tanh    sech    csch    coth
               atanh   asech   acsch   acoth

       cannot be computed for all arguments because that would mean dividing by zero or taking logarithm of
       zero. These situations cause fatal runtime errors looking like this

               cot(0): Division by zero.
               (Because in the definition of cot(0), the divisor sin(0) is 0)
               Died at ...

       or

               atanh(-1): Logarithm of zero.
               Died at...

       For the "csc", "cot", "asec", "acsc", "acot", "csch", "coth", "asech", "acsch", the argument cannot
       be 0 (zero).  For the logarithmic functions and the "atanh", "acoth", the argument cannot be 1 (one).
       For the "atanh", "acoth", the argument cannot be "-1" (minus one).  For the "atan", "acot", the
       argument cannot be "i" (the imaginary unit).  For the "atan", "acoth", the argument cannot be "-i"
       (the negative imaginary unit).  For the "tan", "sec", "tanh", the argument cannot be pi/2 + k * pi,
       where k is any integer.  atan2(0, 0) is undefined, and if the complex arguments are used for atan2(),
       a division by zero will happen if z1**2+z2**2 == 0.

       Note that because we are operating on approximations of real numbers, these errors can happen when
       merely `too close' to the singularities listed above.

ERRORS DUE TO INDIGESTIBLE ARGUMENTS
       The "make" and "emake" accept both real and complex arguments.  When they cannot recognize the
       arguments they will die with error messages like the following

           Math::Complex::make: Cannot take real part of ...
           Math::Complex::make: Cannot take real part of ...
           Math::Complex::emake: Cannot take rho of ...
           Math::Complex::emake: Cannot take theta of ...

BUGS
       Saying "use Math::Complex;" exports many mathematical routines in the caller environment and even
       overrides some ("sqrt", "log", "atan2").  This is construed as a feature by the Authors, actually...
       ;-)

       All routines expect to be given real or complex numbers. Don't attempt to use BigFloat, since Perl
       has currently no rule to disambiguate a '+' operation (for instance) between two overloaded entities.

       In Cray UNICOS there is some strange numerical instability that results in root(), cos(), sin(),
       cosh(), sinh(), losing accuracy fast.  Beware.  The bug may be in UNICOS math libs, in UNICOS C
       compiler, in Math::Complex.  Whatever it is, it does not manifest itself anywhere else where Perl
       runs.

SEE ALSO
       Math::Trig

AUTHORS
       Daniel S. Lewart <lewart!at!uiuc.edu> Jarkko Hietaniemi <jhi!at!iki.fi> Raphael Manfredi
       <Raphael_Manfredi!at!pobox.com>

LICENSE
       This library is free software; you can redistribute it and/or modify it under the same terms as Perl
       itself.



perl v5.12.5                                     2012-11-03                               Math::Complex(3pm)