(octave.info)Complex Arithmetic

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17.2 Complex Arithmetic
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In the descriptions of the following functions, Z is the complex number
X + IY, where I is defined as ‘sqrt (-1)’.

 -- : abs (Z)
     Compute the magnitude of Z.

     The magnitude is defined as |Z| = ‘sqrt (x^2 + y^2)’.

     For example:

          abs (3 + 4i)
               ⇒ 5

     See also: Note: arg.

 -- : arg (Z)
 -- : angle (Z)
     Compute the argument, i.e., angle of Z.

     This is defined as, THETA = ‘atan2 (Y, X)’, in radians.

     For example:

          arg (3 + 4i)
               ⇒ 0.92730

     See also: Note: abs.

 -- : conj (Z)
     Return the complex conjugate of Z.

     The complex conjugate is defined as ‘conj (Z)’ = X - IY.

     See also: Note: real, Note: imag.

 -- : cplxpair (Z)
 -- : cplxpair (Z, TOL)
 -- : cplxpair (Z, TOL, DIM)
     Sort the numbers Z into complex conjugate pairs ordered by
     increasing real part.

     The negative imaginary complex numbers are placed first within each
     pair.  All real numbers (those with ‘abs (imag (Z)) / abs (Z) <
     TOL’) are placed after the complex pairs.

     TOL is a weighting factor in the range [0, 1) which determines the
     tolerance of the matching.  The default value is ‘100 * eps’ and
     the resulting tolerance for a given complex pair is ‘TOL * abs
     (Z(i)))’.

     By default the complex pairs are sorted along the first
     non-singleton dimension of Z.  If DIM is specified, then the
     complex pairs are sorted along this dimension.

     Signal an error if some complex numbers could not be paired.
     Signal an error if all complex numbers are not exact conjugates (to
     within TOL).  Note that there is no defined order for pairs with
     identical real parts but differing imaginary parts.

          cplxpair (exp (2i*pi*[0:4]'/5)) == exp (2i*pi*[3; 2; 4; 1; 0]/5)

 -- : imag (Z)
     Return the imaginary part of Z as a real number.

     See also: Note: real, Note: conj.

 -- : real (Z)
     Return the real part of Z.

     See also: Note: imag, Note: conj.