Math::Complex(3) Perl Programmers Reference GuideMath::Complex(3)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
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
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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 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 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 = <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)16/Sep/1999 perl 5.005, patch 03 4
Math::Complex(3) Perl Programmers Reference GuideMath::Complex(3)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 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.
Getting the number mathematicians call j such that:
1 + j + j*j = 0;
is a simple matter of writing:
$j = ((root(1, 3))[1];
The kth root for z = [r,t] is given by:
(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
The spaceship comparison operator, <=>, is also defined.
In order to ensure its restriction to real numbers is
conform to what you would expect, the comparison is run on
the real part of the complex number first, and imaginary
parts are compared only when the real parts match.
CREATION
To create a complex number, use either:
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$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 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";
$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...
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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 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.
Note that because we are operating on approximations of
real numbers, these errors can happen when merely `too
close' to the singularities listed above. For example
tan(2*atan2(1,1)+1e-15) will die of division by zero.
ERRORS DUE TO INDIGESTIBLE ARGUMENTS
The make and emake accept both real and complex arguments.
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
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.
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
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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