Complex Numbers

Representation methods

The methods are:

• Cartesian representation: $z=x+iy$
• Polar representation: $z=pe^{i\theta}$

Here:

• $x = p\cos{\theta}$ - real part
• $y=p\sin{\theta}$ - imaginary part
• $p=\sqrt{x^{2} + y^{2}}$ - modulus
• $\theta=\tan^{-1}{(\frac{y}{x})}$ - arg angle

Euler’s Formula

For $x\in\mathbb{R}$: $$

$$e^{ix} = \cos{x} + i\sin{x}$$

Proof Hint

Use Taylor series for $e^x,\cos{x}, \sin{x}$. $$

Euler’s Identity

One of the most beautiful equations in mathematics.

$$e^{i\pi} + 1 = 0$$