Math::Complex(3) Math::Complex(3)
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
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Math::Complex(3) Math::Complex(3)
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 will be noted 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 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
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Math::Complex(3) Math::Complex(3)
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.
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 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
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Math::Complex(3) Math::Complex(3)
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 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)
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))
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Math::Complex(3) Math::Complex(3)
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 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 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. 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:
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Math::Complex(3) Math::Complex(3)
$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).
STRINGIFICATION
When printed, a complex number is usually shown under its cartesian form
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 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.
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 numbers
is simple and almost transparent.
Here are some examples:
use Math::Complex;
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Math::Complex(3) Math::Complex(3)
$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 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 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, sech, the argument cannot be pi/2 + k * pi, where
k is any integer.
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Math::Complex(3) Math::Complex(3)
BUGS
Saying use Math::Complex; exports many mathematical routines in the
caller 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 to
use BigFloat, since Perl has currently no rule to disambiguate a '+'
operation (for instance) between two overloaded entities.
AUTHORS
Raphael Manfredi <Raphael_Manfredi@grenoble.hp.com> and Jarkko Hietaniemi
<jhi@iki.fi>.
Extensive patches by Daniel S. Lewart <d-lewart@uiuc.edu>.
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