- #1
Mandelbroth
- 611
- 24
The other day, I was thinking about Fourier series. Because eix is periodic, with period of 2 pi, we can use the Fourier series...
[itex]\displaystyle \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{ix} \ dx + \frac{1}{\pi}\sum_{n=1}^{\infty}\left[\left(\int_{-\pi}^{\pi} e^{ix} cos(nx) \ dx \right) cos(nx) + \left(\int_{-\pi}^{\pi} e^{ix} sin(nx) \ dx\right) sin(nx)\right] = \frac{0}{2\pi} + \frac{1}{\pi}(\pi cos{x} + i\pi sin{x}) = cos{x} + i sin{x}[/itex]
...to describe eix, right?
Isn't this an easier way to prove Euler's formula than using Taylor expansion? Or...am I missing something?
[itex]\displaystyle \frac{1}{2\pi} \int_{-\pi}^{\pi} e^{ix} \ dx + \frac{1}{\pi}\sum_{n=1}^{\infty}\left[\left(\int_{-\pi}^{\pi} e^{ix} cos(nx) \ dx \right) cos(nx) + \left(\int_{-\pi}^{\pi} e^{ix} sin(nx) \ dx\right) sin(nx)\right] = \frac{0}{2\pi} + \frac{1}{\pi}(\pi cos{x} + i\pi sin{x}) = cos{x} + i sin{x}[/itex]
...to describe eix, right?
Isn't this an easier way to prove Euler's formula than using Taylor expansion? Or...am I missing something?