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Math::Complex(3)       Perl Programmers	Reference Guide	      Math::Complex(3)

       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

	       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

       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

	       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)


	       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

	       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.

       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

       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


       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;

	       $j = (root(1, 3))[1];
	       print "j	= $j\n";	       # Prints	"j = [1,2pi/3]"
	       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".

       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.


       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.

       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;

       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


       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 ...


	       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.


       Daniel S. Lewart	<lewart!at!>, Jarkko Hietaniemi
       <jhi!at!>,	Raphael	Manfredi <Raphael_Manfredi!at!>,
       Zefram <>

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

perl v5.26.0			  2017-04-19		      Math::Complex(3)


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