So if I have a function like the following, where $f$ is a complex function and B is real: $$S(\lambda) = 1 + iBf(\lambda)$$ Suppose that I know that its modulus must be equal to 1, therefore:
Now how do I determine the conditions for the real part and its imaginary part? I know that: $$|1 + iBf(\lambda)|^2 = 1$$
From here: $$|1 + iB [Re(f(\lambda)) + iIm(f(\lambda)]|^2 = 1$$
Then I can say that the imaginary and real parts of the function would be: $$Im(S(\lambda)) = 1+ BRe(f(\lambda))$$ $$Re(S(\lambda)) = -BIm(f(\lambda))$$
Is my reasoning correct?