Can someone tell me if this is correct... if not, what would be the correct absolute value of the imaginary number below ?
Where $i$ is the imaginary number, and $\lambda$ is some constant, I have this equation. $$|\frac{g^{t}}{g^{t-1}}|=|\frac{1}{1-2i\lambda sin(\Delta x)}|$$ Since this is in the real number field, these have recipricals. So, does that mean :
$$|\frac{g^{t-1}}{g^{t}}|=|1-2i\lambda sin(\Delta x)|=\sqrt{1+4\lambda^2}\\ |\frac{g^t}{g^{t-1}}|=\frac{1}{\sqrt{1+4\lambda^2}}$$
Or is $|\frac{g^{t}}{g^{t-1}}|*|\frac{g^{t-1}}{g^{t}}|\neq1$?
For all complex numbers $z_1, z_2$ we have that $$|z_1 z_2|=|z_1||z_2|$$ Or equivalently $$\bigg|\frac{z_1}{z_2}\bigg| =\frac{|z_1|}{|z_2|}$$ So what you have written is true. But if the value of $\Delta x$ is also complex then you have incorrectly implied that $|\sin{(\Delta x)}|=1$ because it is possible for $\sin{(\Delta x)}$ to take any complex value for complex arguments.