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### Contents

```
Math::Complex(3)					      Math::Complex(3)

```

### NAME[Toc][Back]

```     Math::Complex - complex numbers and associated mathematical functions
```

### SYNOPSIS[Toc][Back]

```	     use Math::Complex;

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

```

### DESCRIPTION[Toc][Back]

```     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

Page 1

Math::Complex(3)					      Math::Complex(3)

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	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 nonnegative
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

Page 2

Math::Complex(3)					      Math::Complex(3)

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.

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

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

Page 3

Math::Complex(3)					      Math::Complex(3)

norm.
```

### OPERATIONS[Toc][Back]

```     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)
~z1 = a - bi
abs(z1) = r1 = sqrt(a*a + b*b)
sqrt(z1) =	sqrt(r1) * exp(i * t1/2)
exp(z1) = exp(a) *	exp(i *	b)
log(z1) = log(r1) + i*t1
sin(z1) = 1/2i (exp(i * z1) - exp(-i * z1))
cos(z1) = 1/2 (exp(i * z1)	+ exp(-i * z1))
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

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))

Page 4

Math::Complex(3)					      Math::Complex(3)

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))

log, csc, cot, acsc, acot,	csch, coth, acosech, acotanh, have aliases ln,
cosec, cotan, acosec, acotan, cosech, cotanh, acosech, acotanh,
respectively.

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));

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[Toc][Back]

```     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:

Page 5

Math::Complex(3)					      Math::Complex(3)

\$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).
```

### STRINGIFICATION[Toc][Back]

```     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;

Math::Complex::display_format('polar');
\$j	= ((root(1, 3));
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[Toc][Back]

```     Thanks to overloading, the	handling of arithmetics	with complex numbers
is	simple and almost transparent.

Here are some examples:

use Math::Complex;

Page 6

Math::Complex(3)					      Math::Complex(3)

\$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";

```

### ERRORS DUE TO DIVISION BY ZERO[Toc][Back]

```     The division (/) and the following	functions

tan
sec
csc
cot
asec
acsc
atan
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 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, sech, the argument	cannot be pi/2 + k * pi, where
k is any integer.

Page 7

Math::Complex(3)					      Math::Complex(3)

```

### BUGS[Toc][Back]

```     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.
```

### AUTHORS[Toc][Back]

```     Raphael Manfredi <Raphael_Manfredi@grenoble.hp.com> and Jarkko Hietaniemi
<jhi@iki.fi>.

Extensive patches by Daniel S. Lewart <d-lewart@uiuc.edu>.

PPPPaaaaggggeeee 8888```
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