Euler's Formula/Examples/e^2 k i pi

Example of Use of Euler's Formula

$\forall k \in \Z: e^{2 k i \pi} = 1$

Proof

\(\ds e^{2 k i \pi}\)

\(=\)

\(\ds \cos 2 k \pi + i \sin 2 k \pi\)

Euler's Formula

\(\ds \)

\(=\)

\(\ds 1 + i \times 0\)

Cosine of Multiple of $\pi$, Sine of Multiple of $\pi$

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations