Math::Complex(3p) Perl Programmers Reference Guide Math::Complex(3p) #
Math::Complex(3p) Perl Programmers Reference Guide Math::Complex(3p)
NNAAMMEE #
Math::Complex - complex numbers and associated mathematical functions
SSYYNNOOPPSSIISS #
use Math::Complex;
$z = Math::Complex->make(5, 6);
$t = 4 - 3*i + $z;
$j = cplxe(1, 2*pi/3);
DDEESSCCRRIIPPTTIIOONN #
This package lets you create and manipulate complex numbers. By default,
_P_e_r_l 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 _i_m_a_g_i_n_a_r_y 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 _r_e_a_l part and "b" is the _i_m_a_g_i_n_a_r_y 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 _c_o_m_p_l_e_x _p_l_a_n_e, 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 _r_h_o the _n_o_r_m or _m_o_d_u_l_u_s and _t_h_e_t_a the _a_r_g_u_m_e_n_t of the
complex number. The _n_o_r_m of "z" is marked here as 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 _y or _t_h_e_t_a is
zero or _p_i.
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
_e_x_t_e_n_s_i_o_n_s 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 RR++ to RR++). If we allow
it to return a complex number, then it can be extended to negative real
numbers to become an application from RR to CC (the set of complex
numbers):
sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
It can also be extended to be an application from CC to CC, whilst its
restriction to RR 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 _a_s _u_s_u_a_l
when the imaginary part is zero (otherwise, it would not be called an
extension, would it?).
A _n_e_w operation possible on a complex number that is the identity for
real numbers is called the _c_o_n_j_u_g_a_t_e, 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:
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 _p_o_s_t_e_r_i_o_r_i our use of the "abs"
notation for the norm.
OOPPEERRAATTIIOONNSS #
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(y, x) = atan(y / x) # Minding the right quadrant, note the order.
The definition used for complex arguments of aattaann22(()) is
-i log((x + iy)/sqrt(x*x+y*y))
Note that atan2(0, 0) is not well-defined.
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))
_a_r_g, _a_b_s, _l_o_g, _c_s_c, _c_o_t, _a_c_s_c, _a_c_o_t, _c_s_c_h, _c_o_t_h, _a_c_o_s_e_c_h, _a_c_o_t_a_n_h, have
aliases _r_h_o, _t_h_e_t_a, _l_n, _c_o_s_e_c, _c_o_t_a_n, _a_c_o_s_e_c, _a_c_o_t_a_n, _c_o_s_e_c_h, _c_o_t_a_n_h,
_a_c_o_s_e_c_h, _a_c_o_t_a_n_h, respectively. "Re", "Im", "arg", "abs", "rho", and
"theta" can be used also as mutators. The "cbrt" returns only one of the
solutions: if you want all three, use the "root" function.
The _r_o_o_t 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)
You can return the _kth root directly by "root(z, n, k)", indexing
starting from _z_e_r_o and ending at _n _- _1.
The _s_p_a_c_e_s_h_i_p numeric 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.
CCRREEAATTIIOONN #
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 the
"make", "emake", "cplx", and "cplxe": the appropriate component of the
argument will be used.
$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 components 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).
DDIISSPPLLAAYYIINNGG #
When printed, a complex number is usually shown under its cartesian style
_a_+_b_i, 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 _s_t_r_i_n_g_i_f_i_c_a_t_i_o_n.
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 "cartesian". Not supplying any argument returns
the current 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 _t_h_i_s 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_*_p_i_/_n (where _n is a
positive integer and _k an integer within [-9, +9]), this is called _p_o_l_a_r
_p_r_e_t_t_y_-_p_r_i_n_t_i_n_g.
For the reverse of stringifying, see the "make" and "emake".
CCHHAANNGGEEDD IINN PPEERRLL 55..66 #
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 sspprriinnttff(())-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 parameter 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.
UUSSAAGGEE #
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.
CCOONNSSTTAANNTTSS #
PPII #
The constant "pi" and some handy multiples of it (pi2, pi4, and pip2
(pi/2) and pip4 (pi/4)) are also available if separately exported:
use Math::Complex ':pi';
$third_of_circle = pi2 / 3;
IInnff The floating point infinity can be exported as a subroutine IInnff(()):
use Math::Complex qw(Inf sinh);
my $AlsoInf = Inf() + 42;
my $AnotherInf = sinh(1e42);
print "$AlsoInf is $AnotherInf\n" if $AlsoInf == $AnotherInf;
Note that the stringified form of infinity varies between platforms: it
can be for example any of
inf
infinity
INF #
1.#INF #
or it can be something else.
Also note that in some platforms trying to use the infinity in arithmetic
operations may result in Perl crashing because using an infinity causes
SIGFPE or its moral equivalent to be sent. The way to ignore this is
local $SIG{FPE} = sub { };
EERRRROORRSS DDUUEE TTOO DDIIVVIISSIIOONN BBYY ZZEERROO OORR LLOOGGAARRIITTHHMM OOFF ZZEERROO #
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 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 _p_i_/_2 _+ _k _*
_p_i, where _k is any integer. atan2(0, 0) is undefined, and if the complex
arguments are used for aattaann22(()), a division by zero will happen if
z1**2+z2**2 == 0.
Note that because we are operating on approximations of real numbers,
these errors can happen when merely `too close' to the singularities
listed above.
EERRRROORRSS DDUUEE TTOO IINNDDIIGGEESSTTIIBBLLEE AARRGGUUMMEENNTTSS #
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 ...
BBUUGGSS #
Saying "use Math::Complex;" exports many mathematical routines 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 rroooott(()), ccooss(()), ssiinn(()), ccoosshh(()), ssiinnhh(()), 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.
SSEEEE AALLSSOO #
Math::Trig
AAUUTTHHOORRSS #
Daniel S. Lewart <_l_e_w_a_r_t_!_a_t_!_u_i_u_c_._e_d_u>, Jarkko Hietaniemi <_j_h_i_!_a_t_!_i_k_i_._f_i>,
Raphael Manfredi <_R_a_p_h_a_e_l___M_a_n_f_r_e_d_i_!_a_t_!_p_o_b_o_x_._c_o_m>, Zefram
<zefram@fysh.org>
LLIICCEENNSSEE #
This library is free software; you can redistribute it and/or modify it
under the same terms as Perl itself.
perl v5.36.3 2023-02-15 Math::Complex(3p)