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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:
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(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 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
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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.
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
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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 =
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)
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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 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
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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).
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;
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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;
$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
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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.
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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