Math::Complex(3perl)

📦

version 5.44.0 · 14 versions

💖 Math::Complex(3perl) 💖

Perl Programmers Reference Guide — bestie edition, no cap ✨

✨ NAME ✨

Math::Complex — complex numbers and associated mathematical functions 💅

Okay so this whole package is your backstage pass to complex numbers: making them, doing math on them, all of it. By default Perl stays in its real-number lane, but one extra use statement unlocks full complex support, along with a full set of mathematical functions typically associated with and/or extended to complex numbers. Slay. Periodt.

🎀 SYNOPSIS 🎀

        use Math::Complex;

        $z = Math::Complex->make(5, 6);
        $t = 4 - 3*i + $z;
        $j = cplxe(1, 2*pi/3);

That's the vibe. Now let's get into it, fr fr.

🦋 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. Iconic behavior honestly.

If you're wondering what complex numbers even are, bestie: they were invented to solve this 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.

Arithmetic with pure imaginary numbers works just like you'd expect 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, the way that this next part just works:

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)

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. Fr fr.

💅 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(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. Not me trying to define it anyway 💀.

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 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. Lowkey handy.

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. Highkey clever.

🌸 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. Periodt. 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). The way that it just quietly fixes itself? Main character energy.

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). Obsessed, honestly.

✨ DISPLAYING ✨

When a complex number gets printed, it usually shows up in its cartesian fit, like a+bi — but bestie, sometimes the polar look [r,t] is just more of a moment. That whole glow-up of turning a complex number into a printable string is called stringification 💅.

So here's the tea: call the class method Math::Complex::display_format and hand it either polar or cartesian, and you've overridden the default display style, which is cartesian. Give it no argument at all and it just tells you what the current setting is. Simple.

You can also override the default on a per-number basis by calling the display_format method (not the class method) instead. Same rules: no argument → you get the current display style for that number. Give it something → congratulations, that's the new display style for that particular number 🎀.

Like this, 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 really tries to show off arguments like k*pi/n (where n is a positive integer and k is an integer within [-9, +9]) — that little flex is called polar pretty-printing ✨.

Wanna go the other direction, from string → number? Check the make and emake (they're over in CREATION, iykyk).

💖 CHANGED IN PERL 5.6 💖

The display_format class method and the matching display_format object method can now be called with a parameter hash instead of just a single parameter. Glow up fr.

The old display format style — the one that can be cartesian or polar — can now be changed with the style parameter:

$j->display_format(style => "polar");

And the one-parameter calling convention still totally works, no cap:

$j->display_format("polar");

There are two new display parameters, bestie 👑

The first one is format, which is a sprintf()-style format string used for both numeric parts of the complex number(s). It's kinda 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"

⚠️ Heads up! This also affects the return values of the display_format methods: in list context the whole parameter hash gets returned, instead of just the style parameter value. That's a potential incompatibility with earlier versions if you've been calling the display_format method in list context. Not me discovering this the hard way 😭

The second new display parameter is polar_pretty_print, which you can set to true or false, with the default being true. Wondering what that even means? Peep the previous section, it's all there 🦋

🌸 USAGE 🌸

Thanks to overloading, doing arithmetic with complex numbers is simple and almost transparent. Iconic behaviour, honestly.

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.

💫 CONSTANTS 💫

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. So system-dependent, smh.

😭 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 { };

💀 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 the logarithm of zero. And those 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 ...

atanh(-1): Logarithm of zero.
Died at...

Let's get specific, because yes this matters 👇

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.

⚠️ And note that because we're operating on approximations of real numbers, these errors can happen when you're merely too close to the singularities listed above. So close-ish counts, unfortunately 🙈

💅 ERRORS DUE TO INDIGESTIBLE ARGUMENTS 💅

The make and emake accept both real and complex arguments. But 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, atan2). This is construed as a feature by the Authors, actually... ;-) — king behaviour, honestly 👑

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. Spooky 🌈

🦋 SEE ALSO 🦋

Math::Trig

👑 AUTHORS 👑

Daniel S. Lewart <lewart!at!uiuc.edu>, Jarkko Hietaniemi <jhi!at!iki.fi>, Raphael Manfredi <Raphael_Manfredi!at!pobox.com>, Zefram <zefram@fysh.org>

🌈 LICENSE 🌈

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

And that's a wrap bestie!! 🎀 This was Math::Complex(3perl) from perl v5.44.0, dated 2026-10-17 — you're all caught up, now go stringify something pretty 💖✨

📦 where this page comes from

arch rolling: perl 5.44.0-1 · Artistic-1.0-Perl AND GPL-1.0-or-later
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-11-03: a language model's rewrite of the original, under its terms