Complex Numbers

Representations

Cartesian Form

$z=a+bi$
Real part $Re(z)$
Imaginary part $Im(z)$
Complex Conjugate $\bar{z}$

Polar Form

$z=r(\cos \varphi +i\sin \varphi )$
Modulus $|z|$
Argument $\arg (z)$

Exponential Form

$z=r{e}^{i\varphi }$
Euler’s Formula

Operations

Addition / Subtraction (best in Cartesian)

Multiplication / Division (best in Polar/Exponential)

Powers (De Moivre’s Theorem)

$n$-th Roots of Complex Numbers

Complex Functions

Complex Logarithm

Complex Trigonometric Functions

Handwritten