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

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 Gregory (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 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 of unity:

	  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> line im sin $sin; hold
	  pdl> line re sin $sin
	  pdl> line abs sin $sin

       Sorry, but I didn't yet try to reproduce the diagram in this text. Just run the commands yourself, making sure that you have loaded
       "PDL::Complex" (and "PDL::Graphics::PGPLOT").

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))

       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 v5.14.2							    2012-05-30							      Complex(3pm)