Math::Complex(3) Perl Programmers Reference GuideMath::Complex(3)NAMEMath::Complex - complex numbers and associated mathemati
cal 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 num
bers. 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
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 vec
torial 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 identi
fied 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 nota
tion 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 repre
sentation) is much more handy for performing multiplica
tions and divisions of complex numbers, whilst the carte
sian notation is better suited for additions and subtrac
tions. 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 mean
ing 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 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. The "sqrt" returns only one of the solu
tions: if you want the both, use the "root" function.
All the common mathematical functions defined on real num
bers 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 num
ber, 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 = <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(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
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, respec
tively. "Re", "Im", "arg", "abs", "rho", and "theta" can
be used also also 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 n roots
of some complex, where n is a strictly positive integer.
There are exactly n such roots, returned as a list. Get
ting 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 con
form 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:
$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).
It is also possible to have a complex number as either
argument of either the "make" or "emake": the appropriate
component of the argument will be used.
$z1 = cplx(-2, 1);
$z2 = cplx($z1, 4);
STRINGIFICATION
When printed, a complex number is usually shown under its
cartesian style a+bi, but there are legitimate cases where
the polar style [r,t] is more appropriate.
By calling the class method "Math::Complex::display_for
mat" and supplying either ""polar"" or ""cartesian"" as an
argument, you override the default display style, which is
""cartesian"". Not supplying any argument returns the cur
rent settings.
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
style for this number. Otherwise whatever you specify 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.
CHANGED IN PERL 5.6
The "display_format" class method and the corresponding
"display_format" object method can now be called using a
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 string to be used for both numeric parts of the
complex number(s). The is somewhat 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 param
eter hash will be returned, as opposed to only the style
parameter value. This is a potential incompatibility with
earlier versions if you have been calling the "dis
play_format" method in list context.
The second new display parameter is
""polar_pretty_print"", which can be set to true or false,
the default being true. See the previous section for what
this means.
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";
$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 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 the logarithmic functions and 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 argu
ment cannot be "-i" (the negative imaginary unit). For
the "tan", "sec", "tanh", the argument cannot be pi/2 + k
* pi, where k is any integer.
Note that because we are operating on approximations of
real numbers, these errors can happen when merely `too
close' to the singularities listed above.
ERRORS DUE TO INDIGESTIBLE ARGUMENTS
The "make" and "emake" accept both real and complex argu
ments. When they cannot recognize the arguments they will
die with error messages 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 rou
tines 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.
In Cray UNICOS there is some strange numerical instability
that results in root(), cos(), sin(), cosh(), sinh(), los
ing 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
Raphael Manfredi <Raphael_Manfredi@pobox.com> and Jarkko
Hietaniemi <jhi@iki.fi>.
Extensive patches by Daniel S. Lewart <d-lewart@uiuc.edu>.
2001-02-22 perl v5.6.1 Math::Complex(3)
