SYNOPSIS

DESCRIPTION

OPERATIONS

CREATION

DISPLAYING

USAGE

CONSTANTS

ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO

ERRORS DUE TO INDIGESTIBLE ARGUMENTS

BUGS

SEE ALSO

AUTHORS

LICENSE

Math::Complex − complex numbers and associated mathematical functions

use Math::Complex;

$z = Math::Complex−>make(5, 6);

$t = 4 − 3*i + $z;

$j = cplxe(1, 2*pi/3);

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:

(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"` 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 *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 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 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
posteriori* our use of the `"abs"`
notation for the norm.

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 *atan2()* 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))

*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 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 *k*th
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 *k*th root directly by `"root(z, n,
k)"`, indexing starting from *zero* and ending
at *n − 1*.

The
*spaceship* 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.

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).

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
`"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 *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*.

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
`"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 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.

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.

**PI**

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;

**Inf**

The floating point infinity can be exported as a subroutine
*Inf()*:

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 { };

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 *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.

Note that because we are operating on approximations of real numbers, these errors can happen when merely ‘too close’ to the singularities listed above.

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 ...

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 *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.

Math::Trig

Daniel S.
Lewart <*lewart!at!uiuc.edu*> Jarkko Hietaniemi
<*jhi!at!iki.fi*> Raphael Manfredi
<*Raphael_Manfredi!at!pobox.com*>

This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

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