Logarithmic and Complex operations

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

• $$\left\{ \begin{array}{l} \vert w\vert=\sqrt{u^2+v^2},\;\;\;\;\ang... ...\sqrt{x^2+y^2},\;\;\;\;\angle z=\tan^{-1}\left( y/x \right) \end{array} \right.$$ (471)

• $$w z=(u+jv)(x+jy)=\vert w\vert e^{j\angle w}\;\vert z\vert e^{j\angle z}$$ (472)

  $$\left\vert wz \right\vert =\vert w\vert\vert z\vert,\;\;\;\;\; e^{j\angle w} e^{j\angle z}=e^{j(\angle w+\angle z)}, \;\;\; \;\;\;\; \angle \left( wz \right) =\angle w+\angle z$$ (473)

• $$\frac{w}{z}=\frac{u+jv}{x+jy}=\frac{\vert w\vert e^{j\angle w}}{\vert z\vert e^{j\angle z}}$$ (474)

  $$\left\vert \frac{w}{z} \right\vert =\frac{\vert w\vert}{\vert z\v... ...,\;\;\;\;\; \frac{e^{j\angle w}}{e^{j\angle z}}=e^{j(\angle w-\angle z)},\;\;\; \;\;\; \angle \left( \frac{w}{z} \right)=\angle w-\angle z$$ (475)

• $$\log \;(ab)=\log a+\log b$$ (476)

• $$\log \;(a/b)=\log a-\log b$$ (477)

• $$\log \;(a^n)=n\;\log a$$ (478)

• $$\log \;(a^{-n})=-n\;\log a$$ (479)