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Math::Complex

Section: Perl Programmers Reference Guide (3pm)
Updated: 2001-09-21
Index

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 bringsfull complex support, along with a full set of mathematical functionstypically associated with and/or extended to complex numbers.

If you wonder what complex numbers are, they were invented to be able to solvethe following equation:

        x*x = -1

and by definition, the solution is noted i (engineers use j instead sincei usually denotes an intensity, but the name does not matter). The numberi is a pure imaginary number.

The arithmetics with pure imaginary numbers works just like you would expectit 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 imaginarypart, and are usually noted:

        a + bi

where "a" is the real part and "b" is the imaginary part. Thearithmetic with complex numbers is straightforward. You have tokeep track of the real and the imaginary parts, but otherwise therules 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 wouldbe the vector originating from (0, 0) to (a, b). It follows that the additionof two complex numbers is a vectorial addition.

Since there is a bijection between a point in the 2D plane and a complexnumber (i.e. the mapping is unique and reciprocal), a complex numbercan also be uniquely identified with polar coordinates:

        [rho, theta]

where "rho" is the distance to the origin, and "theta" the angle betweenthe vector and the x axis. There is a notation for this using theexponential form, which is:

        rho * exp(i * theta)

where i is the famous imaginary number introduced above. Conversionbetween 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 yaxes. Mathematicians call rho the norm or modulus and thetathe argument of the complex number. The norm of "z" will benoted abs(z).

The polar notation (also known as the trigonometricrepresentation) is much more handy for performing multiplications anddivisions of complex numbers, whilst the cartesian notation is bettersuited for additions and subtractions. Real numbers are on the xaxis, and therefore theta is zero or pi.

All the common operations that can be performed on a real number havebeen defined to work on complex numbers as well, and are merelyextensions of the operations defined on real numbers. This meansthey keep their natural meaning when there is no imaginary part, providedthe number is within their definition set.

For instance, the "sqrt" routine which computes the square root ofits argument is only defined for non-negative real numbers and yields anon-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 tonegative real numbers to become an application from R to C (theset 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 usingthe following definition:

        sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)

Indeed, a negative real number can be noted "[x,pi]" (the modulusx is always non-negative, so "[x,pi]" is really "-x", a negativenumber) 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 thatare extended to complex numbers share that same property of workingas usual when the imaginary part is zero (otherwise, it would notbe called an extension, would it?).

A new operation possible on a complex number that isthe identity for real numbers is called the conjugate, and is notedwith 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 thedistance 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. standsfor the absolute value). This example explains why the norm of "z" isnoted abs(z): it extends the "abs" function to complex numbers, yetis the regular "abs" we know when the complex number actually has noimaginary 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))

The following extra operations are supported on both real and complexnumbers:

        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 nroots of some complex, where n is a strictly positive integer.There are exactly n such roots, returned as a list. Getting thenumber 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)

You can return the kth root directly by "root(z, n, k)",indexing starting from zero and ending at n - 1.

The spaceship comparison operator, <=>, is also defined. Inorder to ensure its restriction to real numbers is conform to what youwould expect, the comparison is run on the real part of the complexnumber first, and imaginary parts are compared only when the realparts 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 anotation 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 themodulus must be non-negative (it represents the distance to the originin 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 ofthe argument will be used.

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

The "new", "make", "emake", "cplx", and "cplxe" will alsounderstand a single (string) argument of the forms

        2-3i        -3i        [2,3]        [2,-3pi/4]        [2]

in which case the appropriate cartesian and exponential componentswill 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 alsounderstand the case of no arguments: this means plain zero or (0, 0).

DISPLAYING

When printed, a complex number is usually shown under its cartesianstyle a+bi, but there are legitimate cases where the polar style[r,t] is more appropriate. The process of converting the complexnumber into a string that can be displayed is known as stringification.

By calling the class method "Math::Complex::display_format" andsupplying either "polar" or "cartesian" as an argument, youoverride the default display style, which is "cartesian". Notsupplying 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 argumentreturns the current display style for this number. Otherwise whatever youspecify 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 usinga 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 stringto be used for both numeric parts of the complex number(s). The issomewhat 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 hashwill be returned, as opposed to only the style parameter value.This is a potential incompatibility with earlier versions if youhave been calling the "display_format" method in list context.

The second new display parameter is "polar_pretty_print", which canbe set to true or false, the default being true. See the previoussection for what this means.

USAGE

Thanks to overloading, the handling of arithmetics with complex numbersis 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 dividingby zero or taking logarithm of zero. These situations cause fatalruntime 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 thelogarithmic functions and the "atanh", "acoth", the argument cannotbe 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 argumentcannot be "-i" (the negative imaginary unit). For the "tan","sec", "tanh", the argument cannot be pi/2 + k * pi, where kis any integer. atan2(0, 0) is undefined, and if the complex argumentsare 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 singularitieslisted 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 errormessages 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 thecaller 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 touse 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 resultsin 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

Daniel S. Lewart <d-lewartAATTuiuc.edu>

Original authors Raphael Manfredi <Raphael_ManfrediAATTpobox.com> andJarkko Hietaniemi <jhiAATTiki.fi>

Index

NAME
SYNOPSIS
DESCRIPTION
OPERATIONS
CREATION
DISPLAYING
USAGE
ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
ERRORS DUE TO INDIGESTIBLE ARGUMENTS
BUGS
AUTHORS

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