A complex number $a+bi$ as a continued fraction.

Question

I tried to learn about continued fractions of rational and irational numbers, but when it comes to complex numbers i cannot find anything resonable about that. Is there a way to write complex numbers as a finite or infinite continued fraction and what is the algorithm for this ?

Answer 1

You might try continued fractions using a quadratic formula solution with complex roots, then treat the real variable as a complex one. The coefficients could either be real or complex. I have an example of this with real coefficients:

          z = (3)/(-1 + 3/(-1 + 3/(-1 + ...... )))

             The solution would be from the equation:

                 z = 3/(-1 + z) or

                 z^2 - z - 3 = 0     

             This would result in:

                  z = 1/2 + (1/2)*(sqrt 11)i

             or

                  z = 1/2 - (1/2)*(sqrt 11)i

                        

Answer 2

Let $z=\frac {p}{q+\frac {p}{q+...}}$

$z=\frac p{q+z}$

$zq+z^2=p$

$z^2+qz-p=0$

If a complex number $z=x+yi$ is a root of a quadratic equation with real coefficients, then another root is $z^*=x-yi$.

The equation is $(z-x-yi)(z-x+yi)=0$

$z^2-2zx+x^2+y^2=0$

Comparing with $z^2+qz-p=0$

gives $p=-(x^2+y^2)$, $q=-2x$