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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>,
       Zefram <zefram AT fysh.org>
LICENSE
       This library is free software; you can redistribute it and/or modify it
       under the same terms as Perl itself.

perl v5.16.3                      2013-03-04                Math::Complex(3pm)