Fformwla Euler

Oddi ar Wicipedia

Daw enw fformwla Euler ar ôl Leonhard Euler.

Mae fformwla Euler yn nodi fod:


e^{i\theta} = \cos (\theta) + \sin (\theta)i\,


ble mae i yn rif dychmygol sydd yn sgwario i roi − 1.


[golygu] Prawf

Mae hyn yn deillio o ehangiadau Cyfres Taylor sy'n nodi fod:


e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \frac{x^{5}}{5!} + \frac{x^{6}}{6!} +\frac{x^{7}}{7!} + ... + \frac{x^{p}}{p!}


\cos {\theta} = 1 - \frac{\theta^{2}}{2!} + \frac{\theta^{4}}{4!} - \frac{\theta^{6}}{6!} + ... + \frac{(-1)^{p}\theta^{2p}}{(2p)!} + ...


\sin {\theta} = \theta - \frac{\theta^{3}}{3!} + \frac{\theta^{5}}{5!} - \frac{\theta^{7}}{7!} + ... + \frac{(-1)^{p}\theta^{2p+1}}{(2p+1)!} + ...


Wedyn o gyfnewid x = iθ yn ehangiad Cyfres Taylor ar gyfer ex yr ydym yn cael:


e^{i\theta} = 1 + (i\theta) + \frac{(i\theta)^{2}}{2!} + \frac{(i\theta)^{3}}{3!} + \frac{(i\theta)^{4}}{4!} + \frac{(i\theta)^{5}}{5!} + \frac{(i\theta)^{6}}{6!} + \frac{(i\theta)^{7}}{7!} + ...


 = 1 + (i\theta) + \frac{i^{2}\theta^{2}}{2!} + \frac{ii^{2}\theta^{3}}{3!} + \frac{i^{2}i^{2}\theta^{4}}{4!} + \frac{ii^{2}i^{2}\theta^{5}}{5!} + \frac{i^{2}i^{2}i^{2}\theta^{6}}{6!} + \frac{ii^{2}i^{2}i^{2}\theta^{7}}{7!} + ...


 = 1 + (i\theta) - \frac{\theta^{2}}{2!} - i\frac{\theta^{3}}{3!} + \frac{\theta^{4}}{4!} + i\frac{\theta^{5}}{5!} - \frac{\theta^{6}}{6!} - i\frac{\theta^{7}}{7!} + ...


 = \{ 1 - \frac{\theta^{2}}{2!} + \frac{\theta^{4}}{4!} -\frac{\theta^{6}}{6!} + ... \} + i\{\theta - \frac{\theta^{3}}{3!} +\frac{\theta^{5}}{5!} - i\frac{\theta^{7}}{7!} + ... \}


 = \cos {(\theta)} + \sin {(\theta)i} \,

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