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Math::Complex(3p)Perl Programmers Reference GuidMath::Complex(3p)


NAME
       Math::Complex - complex numbers and associated mathemati-
       cal 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 num-
       bers. 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:





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               (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 vec-
       torial 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 identi-
       fied 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 nota-
       tion 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" will be noted abs(z).

       The polar notation (also known as the trigonometric repre-
       sentation) is much more handy for performing multiplica-
       tions and divisions of complex numbers, whilst the carte-
       sian notation is better suited for additions and subtrac-
       tions. Real numbers are on the x axis, and therefore 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



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       meaning when there is no imaginary part, provided the num-
       ber 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 solu-
       tions: if you want the both, use the "root" function.

       All the common mathematical functions defined on real num-
       bers 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



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               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 num-
       ber, 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(z1, z2) = atan(z1/z2)

       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)





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               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, respec-
       tively.  "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" func-
       tion.

       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. Get-
       ting 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)

       The spaceship comparison operator, <=>, is also defined.
       In order to ensure its restriction to real numbers is con-
       form to what you would expect, the comparison is run on



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       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]

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

STRINGIFICATION
       When printed, a complex number is usually shown under its



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       cartesian style a+bi, but there are legitimate cases where
       the polar style [r,t] is more appropriate.

       By calling the class method "Math::Complex::display_for-
       mat" and supplying either "polar" or "cartesian" as an
       argument, you override the default display style, which is
       "cartesian". Not supplying any argument returns the cur-
       rent 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.

       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 for-
       mat string to be used for both numeric parts of the com-
       plex 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



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               $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 param-
       eter 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 "dis-
       play_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.

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 ...



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       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 inte-
       ger.

       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 argu-
       ments.  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 rou-
       tines in the caller environment and even overrides some
       ("sqrt", "log").  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(), los-
       ing 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.

AUTHORS
       Daniel S. Lewart <d-lewart@uiuc.edu>

       Original authors Raphael Manfredi <Raphael_Man-
       fredi@pobox.com> and Jarkko Hietaniemi <jhi@iki.fi>




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