Math::Complex - phpMan

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.