use Math::Complex; $z = Math::Complex->make(5, 6); $t = 4 - 3*i + $z; $j = cplxe(1, 2*pi/3);
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
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 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.
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 substractions. Real numbers are on the x axis, and therefore theta is zero.
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 positive real numbers and yields a positive 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
(the modulus x is always positive, 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.
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 = 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 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: 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.
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)) abs(z1) = r1 atan2(z1, z2) = atan(z1/z2)
The following extra operations are supported on both real and complex numbers:
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) cotan(z) = 1 / tan(z)
asin(z) = -i * log(i*z + sqrt(1-z*z)) acos(z) = -i * log(z + sqrt(z*z-1)) atan(z) = i/2 * log((i+z) / (i-z)) acotan(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) cotanh(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)) acotanh(z) = 1/2 * log((1+z) / (z-1))
The root function is available to compute all the nth 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));
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 spaceshift operation 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.
$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 trigonometric 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
e is used as a notation for complex numbers in the trigonometric 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 positive (it represents the distance to the
origin in the complex plane).
By calling the routine
Math::Complex::display_format and supplying either
"cartesian", you override the default display format, which is
"cartesian". Not supplying any argument returns the current setting.
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.
Math::Complex::display_format('polar'); $j = ((root(1, 3)); 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]).
Here are some examples:
$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";
use Math::Complex;exports many mathematical routines in the caller environment. This is construed as a feature by the Author, actually... ;-)
The code is not optimized for speed, although we try to use the cartesian form for addition-like operators and the trigonometric form for all multiplication-like operators.
arg routine does not ensure the angle is within the range
[-pi,+pi] (a side effect caused by multiplication and division using the
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.