how matlab calculate the sqrt of complex number?

Hi, I've written an algorithm which calculate the sqrt of complex number.

%%%%%%%
a_re=-3;
a_im=-6;

rho= sqrt(a_re*a_re+a_im*a_im);
teta=atan(a_im/a_re);

k=0;
risre=sqrt(rho)* cos(teta/2+2*pi*k/2)
risim=sqrt(rho)* sin(teta/2+2*pi*k/2)
%%%%%%%

but the result is'n the same of (-3-6i)^(1/2). Why? I would like to have the same results.

Answer 1

[Babak]

usually powers of between 0 and 1 of complex numbers have multiple answers! As an example, (-3-6i)^(1/2)

[THETA,RHO] = cart2pol(-3,-6)

results in

 THETA =

    -2.0344

 RHO =

     6.7082

Thus

 -3-6i = RHO*exp(i*THETA)

and therefore its half root is

 (-3-6i)^.5 = RHO^.5*exp(i*THETA/2) = 1.3617 - 2.2032i

and

 (-3-6i)^.5 = RHO^.5*exp(i*(THETA+2*pi)/2) = -1.3617 + 2.2032i

This is while MATLAB's sqrt function only gives the first answer, i.e. RHO^.5*exp(i*THETA/2)