Fformiwla Euler

Mae fformiwla Euler yn nodi fod:

Portread 1753 gan Emanuel Handmann o Leonhard Euler. Mae'n bosibl fod ganddo broblem ar ei lygad dde (strabismus o bosib)..

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

ble mae $i$ yn rhif dychmygol sydd yn sgwario i roi $-1$.

Daw'r enw "fformiwla Euler" ar ôl y mathemategydd Leonhard Euler.

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 $e^{x}$ rydym yn cael:

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

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

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

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

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