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Complex(3)	      User Contributed Perl Documentation	    Complex(3)

NAME
       PDL::Complex - handle complex numbers

SYNOPSIS
	 use PDL;
	 use PDL::Complex;

DESCRIPTION
       This module features a growing number of	functions manipulating complex
       numbers.	These are usually represented as a pair	"[ real	imag ]"	or "[
       angle phase ]". If not explicitly mentioned, the	functions can work
       inplace (not yet	implemented!!!)	and require rectangular	form.

       While there is a	procedural interface available ("$a/$b*$c <=> Cmul
       (Cdiv $a, $b), $c)"), you can also opt to cast your pdl's into the
       "PDL::Complex" datatype,	which works just like your normal piddles, but
       with all	the normal perl	operators overloaded.

       The latter means	that "sin($a) +	$b/$c" will be evaluated using the
       normal rules of complex numbers,	while other pdl	functions (like	"max")
       just treat the piddle as	a real-valued piddle with a lowest dimension
       of size 2, so "max" will	return the maximum of all real and imaginary
       parts, not the "highest"	(for some definition)

TIPS, TRICKS & CAVEATS
       o   "i" is a constant exported by this module, which represents
	   "-1**0.5", i.e. the imaginary unit. it can be used to quickly and
	   conviniently	write complex constants	like this: "4+3*i".

       o   Use "r2C(real-values)" to convert from real to complex, as in "$r =
	   Cpow	$cplx, r2C 2". The overloaded operators	automatically do that
	   for you, all	the other functions, do	not. So	"Croots	1, 5" will
	   return all the fifths roots of 1+1*i	(due to	threading).

       o   use "cplx(real-valued-piddle)" to cast from normal piddles into the
	   complex datatype. Use "real(complex-valued-piddle)" to cast back.
	   This	requires a copy, though.

       o   This	module has received some testing by Vanuxem GrA(C)gory
	   (g.vanuxem at wanadoo dot fr). Please report	any other errors you
	   come	across!

EXAMPLE	WALK-THROUGH
       The complex constant five is equal to "pdl(1,0)":

	  pdl> p $x = r2C 5
	  5 +0i

       Now calculate the three cubic roots of of five:

	  pdl> p $r = Croots $x, 3
	  [1.70998 +0i	-0.854988 +1.48088i  -0.854988 -1.48088i]

       Check that these	really are the roots:

	  pdl> p $r ** 3
	  [5 +0i  5 -1.22465e-15i  5 -7.65714e-15i]

       Duh! Could be better. Now try by	multiplying $r three times with
       itself:

	  pdl> p $r*$r*$r
	  [5 +0i  5 -4.72647e-15i  5 -7.53694e-15i]

       Well... maybe "Cpow" (which is used by the "**" operator) isn't as bad
       as I thought. Now multiply by "i" and negate, which is just a very
       expensive way of	swapping real and imaginary parts.

	  pdl> p -($r*i)
	  [0 -1.70998i	1.48088	+0.854988i  -1.48088 +0.854988i]

       Now plot	the magnitude of (part of) the complex sine. First generate
       the coefficients:

	  pdl> $sin = i	* zeroes(50)->xlinvals(2,4) + zeroes(50)->xlinvals(0,7)

       Now plot	the imaginary part, the	real part and the magnitude of the
       sine into the same diagram:

	  pdl> use PDL::Graphics::Gnuplot
	  pdl> gplot( with => 'lines',
		     PDL::cat(im ( sin $sin ),
			      re ( sin $sin ),
			      abs( sin $sin ) ))

       An ASCII	version	of this	plot looks like	this:

	 30 ++-----+------+------+------+------+------+------+------+------+-----++
	    +	   +	  +	 +	+      +      +	     +	    +	   +	  +
	    |									$$|
	    |								       $  |
	 25 ++								     $$	 ++
	    |								   ***	  |
	    |								 **   *** |
	    |							      $$*	 *|
	 20 ++							     $**	 ++
	    |							  $$$*		 #|
	    |						       $$$   *		# |
	    |						     $$	    *		# |
	 15 ++						  $$$	    *	       # ++
	    |					       $$$	  **	       #  |
	    |					   $$$$		 *	      #	  |
	    |				       $$$$		 *	      #	  |
	 10 ++				  $$$$$			*	     #	 ++
	    |			     $$$$$		       *	     #	  |
	    |		      $$$$$$$			      *		    #	  |
	  5 ++	     $$$############			      *		    #	 ++
	    |*****$$$###	    ###			     *		   #	  |
	    *	 #*****		       #		     *		   #	  |
	    | ###      ***		###		   **		   #	  |
	  0 ##		  ***		   #		  *		  #	 ++
	    |		     *		    #		  *		 #	  |
	    |		      ***	     #		**		 #	  |
	    |			 *	      #	       *		#	  |
	 -5 ++			  **	       #      *			#	 ++
	    |			    ***		##  **		       #	  |
	    |			       *	  #*		      #		  |
	    |				****	***##		     #		  |
	-10 ++				    ****     #		    #		 ++
	    |					      #		    #		  |
	    |					       ##	  ##		  |
	    +	   +	  +	 +	+      +      +	 ### + ###  +	   +	  +
	-15 ++-----+------+------+------+------+------+-----###-----+------+-----++
	    0	   5	  10	 15	20     25     30     35	    40	   45	  50

FUNCTIONS
   cplx	real-valued-pdl
       Cast a real-valued piddle to the	complex	datatype. The first dimension
       of the piddle must be of	size 2.	After this the usual (complex)
       arithmetic operators are	applied	to this	pdl, rather than the normal
       elementwise pdl operators.  Dataflow to the complex parent works. Use
       "sever" on the result if	you don't want this.

   complex real-valued-pdl
       Cast a real-valued piddle to the	complex	datatype without dataflow and
       inplace.	Achieved by merely reblessing a	piddle.	The first dimension of
       the piddle must be of size 2.

   real	cplx-valued-pdl
       Cast a complex valued pdl back to the "normal" pdl datatype. Afterwards
       the normal elementwise pdl operators are	used in	operations. Dataflow
       to the real parent works. Use "sever" on	the result if you don't	want
       this.

   r2C
	 Signature: (r(); [o]c(m=2))

       convert real to complex,	assuming an imaginary part of zero

       r2C does	not process bad	values.	 It will set the bad-value flag	of all
       output piddles if the flag is set for any of the	input piddles.

   i2C
	 Signature: (r(); [o]c(m=2))

       convert imaginary to complex, assuming a	real part of zero

       i2C does	not process bad	values.	 It will set the bad-value flag	of all
       output piddles if the flag is set for any of the	input piddles.

   Cr2p
	 Signature: (r(m=2); float+ [o]p(m=2))

       convert complex numbers in rectangular form to polar (mod,arg) form.
       Works inplace

       Cr2p does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cp2r
	 Signature: (r(m=2); [o]p(m=2))

       convert complex numbers in polar	(mod,arg) form to rectangular form.
       Works inplace

       Cp2r does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cmul
	 Signature: (a(m=2); b(m=2); [o]c(m=2))

       complex multiplication

       Cmul does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cprodover
	 Signature: (a(m=2,n); [o]c(m=2))

       Project via product to N-1 dimension

       Cprodover does not process bad values.  It will set the bad-value flag
       of all output piddles if	the flag is set	for any	of the input piddles.

   Cscale
	 Signature: (a(m=2); b(); [o]c(m=2))

       mixed complex/real multiplication

       Cscale does not process bad values.  It will set	the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cdiv
	 Signature: (a(m=2); b(m=2); [o]c(m=2))

       complex division

       Cdiv does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Ccmp
	 Signature: (a(m=2); b(m=2); [o]c())

       Complex comparison oeprator (spaceship).	It orders by real first, then
       by imaginary. Hm, but it	is mathematical	nonsense! Complex numbers
       cannot be ordered.

       Ccmp does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cconj
	 Signature: (a(m=2); [o]c(m=2))

       complex conjugation. Works inplace

       Cconj does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cabs
	 Signature: (a(m=2); [o]c())

       complex "abs()" (also known as modulus)

       Cabs does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cabs2
	 Signature: (a(m=2); [o]c())

       complex squared "abs()" (also known squared modulus)

       Cabs2 does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Carg
	 Signature: (a(m=2); [o]c())

       complex argument	function ("angle")

       Carg does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Csin
	 Signature: (a(m=2); [o]c(m=2))

	 sin (a) = 1/(2*i) * (exp (a*i)	- exp (-a*i)). Works inplace

       Csin does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Ccos
	 Signature: (a(m=2); [o]c(m=2))

	 cos (a) = 1/2 * (exp (a*i) + exp (-a*i)). Works inplace

       Ccos does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Ctan	a [not inplace]
	 tan (a) = -i *	(exp (a*i) - exp (-a*i)) / (exp	(a*i) +	exp (-a*i))

   Cexp
	 Signature: (a(m=2); [o]c(m=2))

       exp (a) = exp (real (a))	* (cos (imag (a)) + i *	sin (imag (a))). Works
       inplace

       Cexp does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Clog
	 Signature: (a(m=2); [o]c(m=2))

       log (a) = log (cabs (a))	+ i * carg (a).	Works inplace

       Clog does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cpow
	 Signature: (a(m=2); b(m=2); [o]c(m=2))

       complex "pow()" ("**"-operator)

       Cpow does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Csqrt
	 Signature: (a(m=2); [o]c(m=2))

       Works inplace

       Csqrt does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Casin
	 Signature: (a(m=2); [o]c(m=2))

       Works inplace

       Casin does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cacos
	 Signature: (a(m=2); [o]c(m=2))

       Works inplace

       Cacos does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Catan cplx [not inplace]
       Return the complex "atan()".

   Csinh
	 Signature: (a(m=2); [o]c(m=2))

	 sinh (a) = (exp (a) - exp (-a)) / 2. Works inplace

       Csinh does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Ccosh
	 Signature: (a(m=2); [o]c(m=2))

	 cosh (a) = (exp (a) + exp (-a)) / 2. Works inplace

       Ccosh does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Ctanh
	 Signature: (a(m=2); [o]c(m=2))

       Works inplace

       Ctanh does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Casinh
	 Signature: (a(m=2); [o]c(m=2))

       Works inplace

       Casinh does not process bad values.  It will set	the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cacosh
	 Signature: (a(m=2); [o]c(m=2))

       Works inplace

       Cacosh does not process bad values.  It will set	the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Catanh
	 Signature: (a(m=2); [o]c(m=2))

       Works inplace

       Catanh does not process bad values.  It will set	the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Cproj
	 Signature: (a(m=2); [o]c(m=2))

       compute the projection of a complex number to the riemann sphere. Works
       inplace

       Cproj does not process bad values.  It will set the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   Croots
	 Signature: (a(m=2); [o]c(m=2,n); int n	=> n)

       Compute the "n" roots of	"a". "n" must be a positive integer. The
       result will always be a complex type!

       Croots does not process bad values.  It will set	the bad-value flag of
       all output piddles if the flag is set for any of	the input piddles.

   re cplx, im cplx
       Return the real or imaginary part of the	complex	number(s) given. These
       are slicing operators, so data flow works. The real and imaginary parts
       are returned as piddles (ref eq PDL).

   rCpolynomial
	 Signature: (coeffs(n);	x(c=2,m); [o]out(c=2,m))

       evaluate	the polynomial with (real) coefficients	"coeffs" at the
       (complex) position(s) "x". "coeffs[0]" is the constant term.

       rCpolynomial does not process bad values.  It will set the bad-value
       flag of all output piddles if the flag is set for any of	the input
       piddles.

AUTHOR
       Copyright (C) 2000 Marc Lehmann <pcg@goof.com>.	All rights reserved.
       There is	no warranty. You are allowed to	redistribute this software /
       documentation as	described in the file COPYING in the PDL distribution.

SEE ALSO
       perl(1),	PDL.

perl v5.24.1			  2017-07-02			    Complex(3)

NAME | SYNOPSIS | DESCRIPTION | TIPS, TRICKS & CAVEATS | EXAMPLE WALK-THROUGH | FUNCTIONS | AUTHOR | SEE ALSO

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