Math::Complex(3p)Perl Programmers Reference GuidMath::Complex(3p)NAMEMath::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 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 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 multipli-
cations 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,
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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 com-
plex 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 solutions:
if you want the both, use the "root" function.
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 a 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:
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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 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 com-
plex 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(y, x) = atan(y / x) # Minding the right quadrant, note the order.
The definition used for complex arguments of atan2() is
-i log((x + iy)/sqrt(x*x+y*y))
The following extra operations are supported on both real
and complex numbers:
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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, respectively.
"Re", "Im", "arg", "abs", "rho", and "theta" can be used
also as mutators. The "cbrt" returns only one of the solu-
tions: 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. Getting the
number mathematicians call "j" such that:
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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)
You can return the kth root directly by "root(z, n, k)",
indexing starting from zero and ending at n - 1.
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:
$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:
$x = cplxe(-3, pi/4);
but that will be silently converted into "[3,-3pi/4]", since
the modulus must be non-negative (it represents the distance
to the origin in the complex plane).
It is also possible to have a complex number as either argu-
ment of the "make", "emake", "cplx", and "cplxe": the
appropriate component of the argument will be used.
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$z1 = cplx(-2, 1);
$z2 = cplx($z1, 4);
The "new", "make", "emake", "cplx", and "cplxe" will also
understand a single (string) argument of the forms
2-3i
-3i
[2,3]
[2,-3pi/4]
[2]
in which case the appropriate cartesian and exponential com-
ponents will be parsed from the string and used to create
new complex numbers. The imaginary component and the theta,
respectively, will default to zero.
The "new", "make", "emake", "cplx", and "cplxe" will also
understand the case of no arguments: this means plain zero
or (0, 0).
DISPLAYING
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. The process of
converting the complex number into a string that can be
displayed is known as stringification.
By calling the class method "Math::Complex::display_format"
and supplying either "polar" or "cartesian" as an argument,
you override the default display style, which is "carte-
sian". Not supplying any argument returns the current set-
tings.
This default can be overridden on a per-number basis by cal-
ling 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,
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+9]), this is called polar pretty-printing.
For the reverse of stringifying, see the "make" and "emake".
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 "carte-
sian" or "polar", can be changed using the "style" parame-
ter.
$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 parame-
ter 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
"display_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 com-
plex numbers is simple and almost transparent.
Here are some examples:
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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 situa-
tions 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 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 argument 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.
atan2(0, 0) is undefined, and if the complex arguments are
used for atan2(), a division by zero will happen if
z1**2+z2**2 == 0.
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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", "atan2"). 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(), losing
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
Daniel S. Lewart <d-lewart@uiuc.edu>
Original authors Raphael Manfredi
<Raphael_Manfredi@pobox.com> and Jarkko Hietaniemi
<jhi@iki.fi>
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