Forms of a complex number

Form of a complex number:

• polar
• trigonometric
• exponential

Polar

Given the point (a, b) I sharpen the complex number z=a+b\cdot{i} whose module is r and its argument is \alpha, its representation in polar shape it is z=r_\alpha

Example of a polar

z=(-3)+4\cdot{i}

\|z\|=\|(-3)+4\cdot{i}\|=\sqrt{(-3)^2+4^2}=\sqrt{9+16}=\sqrt{25}=5

\alpha=\frac{atan2(4, -3)\cdot{360}}{2\cdot\pi}=\frac{2.2143\cdot{360}}{2\cdot\pi}=\frac{797.148}{2\cdot\pi}=126.8701^{\circ}\approx\frac{2\cdot\pi}{3}

So we have to:

z=5_{\frac{2\cdot\pi}{3}}

Trigonometric

It can also be represented in trigonometric shape where

\begin{cases}a=r\cdot\cos{\alpha} \\ b=r\cdot\sin{\alpha} \end{cases}

with what we have

z=a+b\cdot{i}=r\cdot(\cos{\alpha}+\sin{\alpha}\cdot{i})

Example of a trigonometric

z=(-3)+4\cdot{i}

\|z\|=\|(-3)+4\cdot{i}\|=\sqrt{(-3)^2+4^2}=\sqrt{9+16}=\sqrt{25}=5

\alpha=\frac{atan2(4, -3)\cdot{360}}{2\cdot\pi}=\frac{2.2143\cdot{360}}{2\cdot\pi}=\frac{797.148}{2\cdot\pi}=126.8701^{\circ}\approx\frac{2\cdot\pi}{3}

So we have to:

z=a+b\cdot{i}=5\cdot(\cos(\frac{2\cdot\pi}{3})+\sin(\frac{2\cdot\pi}{3})\cdot{i})

Exponential

It can also be represented in exponentially shape where

\begin{cases}\sin\alpha=\frac{e^{\alpha\cdot{i}}-e^{(-\alpha)\cdot{i}}}{2\cdot{i}} \\ \cos\alpha=\frac{e^{\alpha\cdot{i}}+e^{(-\alpha)\cdot{i}}}{2}\end{cases}

with what we have to

z=a+b\cdot{i}=r\cdot(\cos{\alpha}+\sin{\alpha}\cdot{i})=r\cdot{e^{\alpha\cdot{i}}}

Example of exponential

z=(-3)+4\cdot{i}

\|z\|=\|(-3)+4\cdot{i}\|=\sqrt{(-3)^2+4^2}=\sqrt{9+16}=\sqrt{25}=5

\alpha=\frac{atan2(4, -3)\cdot{360}}{2\cdot\pi}=\frac{2.2143\cdot{360}}{2\cdot\pi}=\frac{797.148}{2\cdot\pi}=126.8701^{\circ}\approx\frac{2\cdot\pi}{3}

So we have to:

z=a+b\cdot{i}=5\cdot{e^{\frac{2\cdot\pi}{3}\cdot{i}}}