Euler Relation

\[ e^{i\theta} = \cos(\theta) + i \sin(\theta) \] \[ e^{-i\theta} = \cos(\theta) - i \sin(\theta) \]
\[\sin(\theta)\] \[cos(\theta)\]
\[ sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2}i \] Para simplificar \(\frac{1}{2i}\), multiplicamos numerador y denominador por \(i\): \[ \frac{1}{2i} = \frac{1}{2i} \cdot \frac{i}{i} = \frac{i}{2i^2} = \frac{i}{2(-1)} = \frac{i}{-2} = -\frac{1}{2}i = \frac{-i}{2} \] \[ \sin(\theta) = \frac{-i}{2} (e^{i\theta} - e^{-i\theta}) \] \[ \sin(\theta) = \frac{1}{2i} (e^{i\theta} - e^{-i\theta}) \] \[ \cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2} \] \[ \cos(\theta) = \frac{1}{2} (e^{i\theta} + e^{-i\theta}) \]
Reemplazando nos queda: \[ e^{i\theta} = \cos(\theta) - i \sin(\theta) \] \[ e^{-i\theta} = \frac{1}{2} (e^{i\theta} + e^{-i\theta}) - i (-\frac{1}{2}i (e^{i\theta} - e^{-i\theta})) \] \[ e^{-i\theta} = \frac{1}{2} (e^{i\theta} + e^{-i\theta}) + \frac{1}{2}i^2 (e^{i\theta} - e^{-i\theta}) \] \[ e^{-i\theta} = \frac{1}{2} (e^{i\theta} + e^{-i\theta}) + \frac{1}{2}(-1) (e^{i\theta} - e^{-i\theta}) \] \[ e^{-i\theta} = \frac{1}{2} (e^{i\theta} + e^{-i\theta}) - \frac{1}{2} (e^{i\theta} - e^{-i\theta}) \] \[ e^{-i\theta} = \frac{1}{2}e^{i\theta} + \frac{1}{2}e^{-i\theta} - \frac{1}{2}e^{i\theta} + \frac{1}{2}e^{-i\theta} \] Cancelamos términos \[ e^{-i\theta} = \color{blue} \frac{1}{2}e^{i\theta} \color{black} + \frac{1}{2}e^{-i\theta} \color{blue} - \frac{1}{2}e^{i\theta} \color{black} + \frac{1}{2}e^{-i\theta} \] \[ e^{-i\theta} = \frac{1}{2}e^{-i\theta} + \frac{1}{2}e^{-i\theta} \] \[ e^{-i\theta} = \frac{e^{-i\theta}}{2} + \frac{e^{-i\theta}}{2} \]
\[ e^{-\infty} = 0 \] \[ e^{+\infty} = \infty \]
\[ e^0 = 1 \]
\[ e^x e^y = e^{x + y} \] \[ \frac{e^x}{e^y} = e^{x - y} \] \[ (e^x)^y = e^{xy} \] \[ \frac{de^x}{dx} = e^x \] \[ \frac{d^n e^{ax}}{dx^n} = a^n e^{ax} \] \[ \sqrt[p]{e^x} = e^{\frac{x}{p}} \] \[ \ln(e^x) = x \] \[ \int e^{ax} dx = \frac{e^{ax}}{a} \]