Logarithmic and Complex operations

Assume $w=u+jv=\vert w\vert\angle w=\vert w\vert e^{j\angle w}$, and $z=x+jy=\vert z\vert\angle z=\vert z\vert e^{j\angle z}$.

•   $$\left\{ \begin{array}{l} \vert w\vert=\sqrt{u^2+v^2},\;\;\;\;\an... ...sqrt{x^2+y^2},\;\;\;\;\angle z=\tan^{-1}\left( y/x \right) \end{array} \right.$$ (115)
•   $$w z=(u+jv)(x+jy)=\vert w\vert e^{j\angle w}\;\vert z\vert e^{j\angle z}$$ (116)
    $$\left\vert wz \right\vert =\vert w\vert\vert z\vert,\;\;\;\;\; e... ...gle z)}, \;\;\;\mbox{or}\;\;\;\; \angle \left( wz \right) =\angle w+\angle z$$ (117)
•   $$\frac{w}{z}=\frac{u+jv}{x+jy}=\frac{\vert w\vert e^{j\angle w}}{\vert z\vert e^{j\angle z}}$$ (118)
    $$\left\vert \frac{w}{z} \right\vert =\frac{\vert w\vert}{\vert z\v... ...z)},\;\;\;\mbox{or}\;\;\; \angle \left( \frac{w}{z} \right)=\angle w-\angle z$$ (119)

•   $$\log \;(ab)=\log a+\log b$$ (120)
•   $$\log \;(a/b)=\log a-\log b$$ (121)
•   $$\log \;(a^n)=n\;\log a$$ (122)
•   $$\log \;(a^{-n})=-n\;\log a$$ (123)