Fformwla Euler

Neidio i: llywio, chwilio
Portread 1753 gan Emanuel Handmann o Leonhard Euler. Mae'n bosibl fod ganddo broblem ar ei lygad dde (strabismus o bosib)..[1]

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 $-1$.

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 $x = i\theta$ yn ehangiad Cyfres Taylor ar gyfer $e^{x}$ yr ydym yn cael:

$e^{i\theta} = 1 + (i\theta) + \frac{(i\theta)^{2}}{2!} + \frac{(i\theta)^{3}}{3!} + \frac{(i\theta)^{4}}{4!} + \frac{(i\theta)^{5}}{5!} + \frac{(i\theta)^{6}}{6!} + \frac{(i\theta)^{7}}{7!} + ...$
$= 1 + (i\theta) + \frac{i^{2}\theta^{2}}{2!} + \frac{ii^{2}\theta^{3}}{3!} + \frac{i^{2}i^{2}\theta^{4}}{4!} + \frac{ii^{2}i^{2}\theta^{5}}{5!} + \frac{i^{2}i^{2}i^{2}\theta^{6}}{6!} + \frac{ii^{2}i^{2}i^{2}\theta^{7}}{7!} + ...$
$= 1 + (i\theta) - \frac{\theta^{2}}{2!} - i\frac{\theta^{3}}{3!} + \frac{\theta^{4}}{4!} + i\frac{\theta^{5}}{5!} - \frac{\theta^{6}}{6!} - i\frac{\theta^{7}}{7!} + ...$
$= \{ 1 - \frac{\theta^{2}}{2!} + \frac{\theta^{4}}{4!} -\frac{\theta^{6}}{6!} + ... \} + i\{\theta - \frac{\theta^{3}}{3!} +\frac{\theta^{5}}{5!} - i\frac{\theta^{7}}{7!} + ... \}$
$= \cos {(\theta)} + \sin {(\theta)i} \,$