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Math::Complex

Section: Perl Programmers Reference Guide (3)
Updated: perl 5.004, patch 04
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.

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 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 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 absnotation 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)        ~z1 = a - bi        abs(z1) = r1 = sqrt(a*a + b*b)        sqrt(z1) = sqrt(r1) * exp(i * t1/2)        exp(z1) = exp(a) * exp(i * b)        log(z1) = log(r1) + i*t1        sin(z1) = 1/2i (exp(i * z1) - exp(-i * z1))        cos(z1) = 1/2 (exp(i * z1) + exp(-i * z1))        atan2(z1, z2) = atan(z1/z2)

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

        Re(z) = a        Im(z) = b        arg(z) = t

        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))

log, csc, cot, acsc, acot, csch, coth,acosech, acotanh, have aliases ln, cosec, cotan,acosec, acotan, cosech, cotanh, acosech, acotanh,respectively.

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)

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 the modulusmust be non-negative (it represents the distance to the origin in the complexplane).

STRINGIFICATION

When printed, a complex number is usually shown under its cartesianform 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 currentsetting.

This default can be overridden on a per-number basis by calling thedisplay_format method instead. As before, not supplying any argumentreturns the current display format for this number. Otherwise whatever youspecify will be the new display format 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 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 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";

ERRORS DUE TO DIVISION BY ZERO

The division (/) and the following functions

        tan        sec        csc        cot        asec        acsc        atan        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 theatanh, acoth, the argument cannot be 1 (one). For theatanh, acoth, the argument cannot be -1 (minus one). For theatan, acot, the argument cannot be i (the imaginary unit).For the atan, acoth, the argument cannot be -i (the negativeimaginary unit). For the tan, sec, tanh, sech, theargument cannot be pi/2 + k * pi, where k is any integer.

BUGS

Saying use Math::Complex; exports many mathematical routines in thecaller 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 touse BigFloat, since Perl has currently no rule to disambiguate a `+'operation (for instance) between two overloaded entities.

AUTHORS

Raphael Manfredi <Raphael_ManfrediAATTgrenoble.hp.com> andJarkko Hietaniemi <jhiAATTiki.fi>.

Extensive patches by Daniel S. Lewart <d-lewartAATTuiuc.edu>.

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