5.6.1 Complex Number System 🔧
| Euler's Formula |
$e^{\pm i\cdot\theta}=\cos(\theta)\pm i\cdot\sin(\theta),$
$\forall\theta\isin\R$
|
| Form Equivalences |
|
| Natural Logarithms of Negatives |
$\ln(-x)=i\cdot\pi+\ln(x),$
$\forall x\gt 0\isin\R$ |
| Natural Logarithms of Imaginaries |
$\ln(i)=\frac{\pi}{2}\cdot i$ |
Proof of Euler's Formula
Let $e^{\pm i\cdot\theta}=\cos(\theta)\pm i\cdot\sin(\theta)$ so that
$$f(\theta)=\frac{\cos(\theta)\pm i\cdot\sin(\theta)}{e^{\pm i\cdot\theta}}$$
Rewrite the exponent
$$f(\theta)=e^{\mp i\cdot\theta}\cdot\big(\cos(\theta)\pm i\cdot\sin(\theta)\big)$$
Derive using the
derivative product rule
Proof of Negative Natural Logarithms
Proof of Imaginary Natural Logarithms
