


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


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:





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


               (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 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



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


       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 an 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



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


       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(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)



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


               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 also 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 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:



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


               $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 either the make or emake: the appropriate
       component of the argument will be used.

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


STRINGIFICATION
       When printed, a complex number is usually shown under its
       cartesian form a+bi, but there are legitimate cases where
       the polar format [r,t] is more appropriate.

       By calling the routine Math::Complex::display_format and
       supplying either "polar" or "cartesian", you override the
       default display format, which is "cartesian". Not
       supplying any argument returns the current setting.

       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 format
       for this number. Otherwise whatever you specify will be
       the new display format for this particular number.

       For instance:

               use Math::Complex;




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


               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 format attempts to emphasize arguments like
       k*pi/n (where n is a positive integer and k an integer
       within [-9,+9]).

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...



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


       For the csc, cot, asec, acsc, acot, csch, coth, asech,
       acsch, the argument cannot be 0 (zero).  For the 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.  For example
       tan(2*atan2(1,1)+1e-15) will die of division by zero.

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 anywhere else
       where Perl runs.

AUTHORS
       Raphael Manfredi <Raphael_Manfredi@grenoble.hp.com> and
       Jarkko Hietaniemi <jhi@iki.fi>.

       Extensive patches by Daniel S. Lewart <d-lewart@uiuc.edu>.








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



























































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