Math::...Complex(3pm)

👑 Math::Cephes::Complex (3pm) 👑

okay bestie, sit down, hydrate, and get comfy, because today we're doing complex numbers and honestly? they ate 💅

This page is filed under the User Contributed Perl Documentation in man section 3pm, and the whole thing is about the module Math::Cephes::Complex.

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✨ NAME ✨

Math::Cephes::Complex — the Perl interface to the cephes complex number routines. that's the name, that's the vibe, no notes.

🌸 SYNOPSIS 🌸

use Math::Cephes::Complex qw(cmplx);
my $z1 = cmplx(2,3);          # $z1 = 2 + 3 i
my $z2 = cmplx(3,4);          # $z2 = 3 + 4 i
my $z3 = $z1->radd($z2);      # $z3 = $z1 + $z2

look at that 🦋 three lines in and you're already doing complex arithmetic like a main character.

🧚 DESCRIPTION 🧚

so like, this module is a layer on top of the basic routines in the cephes math library to handle complex numbers. and there are so many ways to summon one, choose your fighter:

my $f = Math::Cephes::Complex->new(3, 2);   # $f = 3 + 2 i
my $g = new Math::Cephes::Complex(5, 3);    # $g = 5 + 3 i
my $h = cmplx(7, 5);                        # $h = 7 + 5 i

that last one, cmplx, is only available if you import cmplx. and if you pass literally no arguments, like

my $h = cmplx();

then the defaults $z = 0 + 0 i are assumed. no thoughts, head empty, zero plus zero i 😌

The real and imaginary parts of a complex number are represented respectively by

$f->{r}; $f->{i};

or, as methods,

$f->r;  $f->i;

and you can set them like

$f->{r} = 4; $f->{i} = 9;

or, again, as methods,

$f->r(4);   $f->i(9);

and to print the whole thing out:

print $f->as_string;

A summary of the usage is as follows 👇

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🌈 csin: Complex circular sine

SYNOPSIS

# void csin();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->csin;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

If

z = x + iy,

then

w = sin x  cosh y  +  i cos x sinh y.

---

🌈 ccos: Complex circular cosine

SYNOPSIS

# void ccos();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ccos;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

If

z = x + iy,

then

w = cos x  cosh y  -  i sin x sinh y.

---

🌈 ctan: Complex circular tangent

SYNOPSIS

# void ctan();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ctan;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

If

z = x + iy,

then

          sin 2x  +  i sinh 2y
    w  =  --------------------.
           cos 2x  +  cosh 2y

⚠️ On the real axis the denominator is zero at odd multiples of PI/2. The denominator is evaluated by its Taylor series near these points. so like, near the spicy spots, it's handled, but be aware 💀

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🌈 ccot: Complex circular cotangent

SYNOPSIS

# void ccot();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ccot;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

If

z = x + iy,

then

          sin 2x  -  i sinh 2y
    w  =  --------------------.
           cosh 2y  -  cos 2x

⚠️ On the real axis, the denominator has zeros at even multiples of PI/2. Near these points it is evaluated by a Taylor series. she's doing the most, but respectfully ✨

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🌈 casin: Complex circular arc sine

SYNOPSIS

# void casin();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->casin;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

Inverse complex sine:

                                      2
w = -i clog( iz + csqrt( 1 - z ) ).

---

🌈 cacos: Complex circular arc cosine

SYNOPSIS

# void cacos();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->cacos;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

w = arccos z  =  PI/2 - arcsin z.

---

🌈 catan: Complex circular arc tangent

SYNOPSIS

# void catan();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->catan;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

If

z = x + iy,

then

             1       (    2x     )
    Re w  =  - arctan(-----------)  +  k PI
             2       (     2    2)
                     (1 - x  - y )

                  ( 2         2)
             1    (x  +  (y+1) )
    Im w  =  - log(------------)
             4    ( 2         2)
                  (x  +  (y-1) )

Where k is an arbitrary integer.

---

🌈 csinh: Complex hyperbolic sine

SYNOPSIS

# void csinh();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->csinh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

csinh z = (cexp(z) - cexp(-z))/2
        = sinh x * cos y  +  i cosh x * sin y .

---

🌈 casinh: Complex inverse hyperbolic sine

SYNOPSIS

# void casinh();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->casinh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

casinh z = -i casin iz .

---

🌈 ccosh: Complex hyperbolic cosine

SYNOPSIS

# void ccosh();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ccosh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

ccosh(z) = cosh x  cos y + i sinh x sin y .

---

🌈 cacosh: Complex inverse hyperbolic cosine

SYNOPSIS

# void cacosh();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->cacosh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

acosh z = i acos z .

---

🌈 ctanh: Complex hyperbolic tangent

SYNOPSIS

# void ctanh();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ctanh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

tanh z = (sinh 2x  +  i sin 2y) / (cosh 2x + cos 2y) .

---

🌈 catanh: Complex inverse hyperbolic tangent

SYNOPSIS

# void catanh();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->catanh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

Inverse tanh, equal to -i catan (iz);

---

🌈 cpow: Complex power function

SYNOPSIS

# void cpow();
# cmplx a, z, w;

$a = cmplx(5, 6);    # $z = 5 + 6 i
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $a->cpow($z);
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

Raises complex A to the complex Zth power. Definition is per AMS55 # 4.2.8, analytically equivalent to cpow(a,z) = cexp(z clog(a)).

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🌈 cmplx: Complex number arithmetic

SYNOPSIS

# typedef struct {
#     double r;     real part
#     double i;     imaginary part
#    }cmplx;

# cmplx *a, *b, *c;

$a = cmplx(3, 5);   # $a = 3 + 5 i
$b = cmplx(2, 3);   # $b = 2 + 3 i

$c = $a->cadd( $b );  #   c = a + b
$c = $a->csub( $b );  #   c = a - b
$c = $a->cmul( $b );  #   c = a * b
$c = $a->cdiv( $b );  #   c = a / b
$c = $a->cneg;        #   c = -a
$c = $a->cmov;        #   c = a

print $c->{r}, '  ', $c->{i};   # prints real and imaginary parts of $c
print $c->as_string;           # prints $c as Re($c) + i Im($c)

DESCRIPTION

Addition:

c.r  =  b.r + a.r
c.i  =  b.i + a.i

Subtraction:

c.r  =  b.r - a.r
c.i  =  b.i - a.i

Multiplication:

c.r  =  b.r * a.r  -  b.i * a.i
c.i  =  b.r * a.i  +  b.i * a.r

Division:

d    =  a.r * a.r  +  a.i * a.i
c.r  = (b.r * a.r  + b.i * a.i)/d
c.i  = (b.i * a.r  -  b.r * a.i)/d

---

🌈 cabs: Complex absolute value

SYNOPSIS

# double a, cabs();
# cmplx z;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$a = cabs( $z );

DESCRIPTION

If z = x + iy

then

a = sqrt( x**2 + y**2 ).

Overflow and underflow are avoided by testing the magnitudes of x and y before squaring 🛡️ If either is outside half of the floating point full scale range, both are rescaled.

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🌈 csqrt: Complex square root

SYNOPSIS

# void csqrt();
# cmplx z, w;

$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->csqrt;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)

DESCRIPTION

If z = x + iy, r = |z|, then

                              1/2
Im w  =  [ (r - x)/2 ]   ,

Re w  =  y / 2 Im w.

📌 Note that -w is also a square root of z. The root chosen is always in the upper half plane.

Because of the potential for cancellation error in r - x, the result is sharpened by doing a Heron iteration (see sqrt.c) in complex arithmetic.

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💀 BUGS 💀

Please report any to Randy Kobes <randy@theoryx5.uwinnipeg.ca> — bestie is accepting mail, be nice 🫶

👀 SEE ALSO 👀

• For the basic interface to the cephes complex number routines, see Math::Cephes.
• See also Math::Complex for a more extensive interface to complex number routines.

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and that's only part 1, bestie 😭 there is so much more where this came from. don't go anywhere 💖

📜 COPYRIGHT 📜

Okay bestie, legal era incoming, but it's actually cute because it's just people saying "here, take my code, be normal about it" 💖

The C code for the Cephes Math Library is Copyright 1984, 1987, 1989, 2002 by Stephen L. Moshier, and is available at http://www.netlib.org/cephes/. Direct inquiries to 30 Frost Street, Cambridge, MA 02140.

The perl interface is copyright 2000, 2002 by Randy Kobes. This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

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No cap — "free software, same terms as Perl itself" is the whole vibe. Respect the OG authors, keep the notices, and you're golden 🫶

And that's the whole page, bestie!! ✨ This beauty is perl v5.42.3, dated 2026-08-19, and it lives at Math::...Complex(3pm). Go forth and conjugate 💅👑🌈