phpman > man > 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 *k*th 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 *k*th 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.