This is one of them crazy things in mathematics. Kind of thing make you want to change career and be a researcher.

If you input imaginary numbers (multiples of \sqrt{-1}) into the exponential function you get the cosine and the sine function: \[ \exp(ix) = \cos(x) + i\sin(x) \]

I mean I am sure you already had some doubts that sine and cosine were related, but the exponential function? Now that is mad!

• $z\in \mathbb{C}$: A complex number
• $Re\{ z \}$: The real part of $z$.
• $Im\{ z \}$: The imaginary part of $z$.
• $\frac{d}{dx}$: Derivative with respect to $x$

Allows you to deal with derivatives simply in AC circuits.