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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 follow-
       ing 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 num-
       ber.

       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 usu-
       ally noted:

	       a + bi

       where "a" is the real part and "b" is the imaginary part. The arithmetic with complex num-
       bers 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 com-
       plex 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"
       will be noted 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 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, pro-
       vided 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-nega-
       tive, 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 (oth-
       erwise, 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 ori-
       gin, 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" nota-
       tion 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)

	       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)

       The spaceship comparison operator, <=>, is also defined. In order to ensure its restric-
       tion 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-nega-
       tive (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 create 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 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_format" and supplying either "polar"
       or "cartesian" as an argument, you override the default display style, which is "carte-
       sian". 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.

       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.

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 argu-
       ment 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.

       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").  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 any-
       where else where Perl runs.

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

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

perl v5.8.0				    2002-06-01			       Math::Complex(3pm)
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