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

       Math::Complex - complex numbers and associated mathematical functions

               use Math::Complex;

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

       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

               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, 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

               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)


               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.

       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

               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 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.

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

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

       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

       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;

               $j = ((root(1, 3))[1];
               print "j = $j\n";               # Prints "j = [1,2pi/3]
               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]).

       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.

       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 ...


               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 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.

       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 ...

       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.

       Raphael Manfredi <> and Jarkko
       Hietaniemi <>.

       Extensive patches by Daniel S. Lewart <>.

3rd Berkeley Distribution    perl 5.005, patch 02             Math::Complex(3)