% The complex conjugate of a complex number is
% obtained by changing the sign of its imaginary
% component.
%
% If
%
%     z = a + bj,
%
% then the complex conjugate of z is
%
%     zc = a - bj.
%
% The complex conjugate is often written
%
%          _
%     zc = z.
%
% In MATLAB and Octave, the function 'conj(...)'
% returns the complex conjugate of a complex
% number.

z = 3.2 - 4.7j

fprintf('\nThe complex conjugate of z:\n')

conj(z)

% We can also construct the complex conjugate
% directly from the real and imaginary
% components.

real(z) - 1j*imag(z)

fprintf('z       = %f%+fj\n', real(z), imag(z))
fprintf('conj(z) = %f%+fj\n', ...
        real(conj(z)), imag(conj(z)))


% Taking the complex conjugate twice returns the
% original complex number:
%
%     conj(conj(z)) = z.
%
% Thus, conjugation is its own inverse.

fprintf('\nThe conjugate of the conjugate:\n')

conj(conj(z))
z

conj(conj(z)) == z


% The conjugate of a real number is the real
% number itself. If
%
%     z = a + 0j,
%
% then
%
%     conj(z) = z.

fprintf('\nThe conjugate of a real number:\n')

conj(3.8)

% Consequently, a complex number is real if and
% only if
%
%     conj(z) = z.


% Conjugation changes the sign of the imaginary
% component but does not change the real
% component:
%
%     Re(conj(z)) = Re(z)
%
%     Im(conj(z)) = -Im(z).

fprintf('\nReal and imaginary components:\n')

real(conj(z))
real(z)

imag(conj(z))
-imag(z)


% The conjugate of an additive inverse is the
% additive inverse of the conjugate:
%
%     conj(-z) = -conj(z).

fprintf('\nConjugation and the additive inverse:\n')

conj(-z)
-conj(z)


% Now consider two complex numbers.

w = 5.3 + 3.2j
z = 3.2 - 4.7j


% The conjugate of a sum is the sum of the
% conjugates:
%
%     conj(w + z) = conj(w) + conj(z).

fprintf('\nThe conjugate of a sum:\n')

conj(w + z)
conj(w) + conj(z)


% Similarly, the conjugate of a difference is
% the difference of the conjugates:
%
%     conj(w - z) = conj(w) - conj(z).

fprintf('\nThe conjugate of a difference:\n')

conj(w - z)
conj(w) - conj(z)


% The conjugate of a product is the product of
% the conjugates:
%
%     conj(w*z) = conj(w)*conj(z).

fprintf('\nThe conjugate of a product:\n')

conj(w*z)
conj(w)*conj(z)


% A complex number multiplied by its conjugate
% is always a nonnegative real number.
%
% If
%
%     z = a + bj,
%
% then
%
%     z*conj(z)
%
%       = (a + bj)(a - bj)
%
%       = a^2 + b^2
%
%       = |z|^2.
%
% Thus,
%
%     z*conj(z) = |z|^2.

fprintf('\nA number times its conjugate:\n')

z*conj(z)
abs(z)^2

% This property is particularly useful when
% dividing complex numbers because it allows
% us to replace a complex denominator with a
% real denominator.


% A complex number and its conjugate have the
% same absolute value:
%
%     |conj(z)| = |z|.
%
% Geometrically, conjugation reflects a point
% across the real axis, which does not change
% its distance from the origin.

fprintf('\nAbsolute value of the conjugate:\n')

abs(conj(z))
abs(z)


% Conjugation reverses the sign of the argument:
%
%     arg(conj(z)) = -arg(z)    (mod 2*pi).
%
% Geometrically, this again corresponds to
% reflecting the complex number across the real
% axis.

fprintf('\nArgument of the conjugate:\n')

angle(conj(z))
-angle(z)


% The real component of a complex number can be
% obtained by adding the number and its conjugate.
%
% If
%
%     z = a + bj,
%
% then
%
%     z + conj(z)
%
%       = (a + bj) + (a - bj)
%
%       = 2a.
%
% Therefore,
%
%                 z + conj(z)
%     Re(z) =     -------------.
%                       2

fprintf('\nRecovering the real component:\n')

real(z)
(z + conj(z))/2


% Similarly, subtracting the conjugate gives
%
%     z - conj(z)
%
%       = (a + bj) - (a - bj)
%
%       = 2bj.
%
% Therefore,
%
%                      z - conj(z)
%     Im(z)j =         -------------
%                           2
%
% and
%
%                      z - conj(z)
%     Im(z) =          -------------.
%                           2j

fprintf('\nRecovering the imaginary component:\n')

1j*imag(z)
(z - conj(z))/2

% Consequently,
%
%     z + conj(z)
%
% is always real, while
%
%     z - conj(z)
%
% is always purely imaginary.

fprintf('\nSum and difference with the conjugate:\n')

z + conj(z)
z - conj(z)


% The absolute-value formula can also be written
% in terms of the conjugate:
%
%                 ___________
%     |z| =      / z*conj(z).
%               V

fprintf('\nCalculating the absolute value using the conjugate:\n')

abs(z)
sqrt(z*conj(z))


% Finally, conjugation has a simple geometric
% interpretation.
%
% If
%
%     z = a + bj,
%
% then
%
%     conj(z) = a - bj.
%
% Thus, z and conj(z) have:
%
%     - the same real component,
%     - opposite imaginary components,
%     - the same absolute value, and
%     - opposite arguments (modulo 2*pi).
%
% Geometrically, taking the complex conjugate
% reflects a complex number across the real
% axis.