Euler's Formula/Corollary

Corollary to Euler's Formula

Let $z \in \C$ be a complex number.

Then:

$e^{-i z} = \cos z - i \sin z$

where:

$e^{-i z}$ denotes the complex exponential function

$\cos z$ denotes the complex cosine function

$\sin z$ denotes complex sine function

$i$ denotes the imaginary unit.

Corollary

This result is often presented and proved separately for arguments in the real domain:

$e^{-i \theta} = \cos \theta - i \sin \theta$

Proof

\(\ds e^{-i z}\)

\(=\)

\(\ds \cos \paren {-z} + i \sin \paren {-z}\)

Euler's Formula

\(\ds \)

\(=\)

\(\ds \cos z + i \sin \paren {-z}\)

Cosine Function is Even

\(\ds \)

\(=\)

\(\ds \cos z - i \sin z\)

Sine Function is Odd

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.16)$